Let me explain, in order, the airports, cities, bus stations, taxis, and train stations you need to get to in order to arrive at the top of Macchu Picchu from Lima. 

Step 1:  Arrive in Lima, Peru. 
I flew from USA to Lima, Peru (Airport code: LIM).  Lima is the biggest city in the region, and very likely Lima will be your entrance-point into Peru.

Step 2:  Lima to Cusco, Peru. 
My flight arrived in Lima around 11 pm.  I purchased the cheapest direct flight from Lima (LIM) to Cusco (CUZ) which left the next morning around 9 am.  I just hung around the airport until the morning flight to Cusco.

Step 3: Cusco airport to Poroy Train Station.
I arrived in Cusco around 11 am.  My friend Andy was meeting me in Cusco late that night, so I needed to stay in Cusco for a day before going to the train.  If you are looking for the city center, it is called 'Centro Historico.' and a cab ride there will be pretty cheap (all cabs in Peru are really cheap unless they give you a 'tourist price.'  There's an ATM in Cusco airport to get cash for the taxi driver. 
The next morning we woke up around 5 am and took a taxi to Poroy train station.  (Taxi drivers will know where this is)  Poroy station is 25 minutes, 10 kilometers Northwest of the center of Cusco. 

Step 4:  Poroy train station to Aguas Calientes, aka Machupicchu ('Estacion de Machu Picchu').
Because we went to Peru in the off season, we didn't have to buy our train tickets in advance, but if you're headed there during the main season, definitely purchase your tickets in advance or they will probably be sold out.  These tickets also somewhat function like airline tickets, as in, the same seat gets more expensive the closer you get to the travel date and as the availability runs low.  The tickets I got were more expensive than normal, about $140 USD round trip, not cheap..  The train goes about 40 miles to Aguas Calientes, but takes 3 hours because it has to weave through the mountains.  Everything in Aguas Calientes is in walking distance, in fact, it has to be, because there are no cars.  The only way for tourists to get to Aguas Calientes is by the train, or hiking the Inca Trail or some other trail (requires permits and guides 🙁   )  When we arrived in Aguas Calientes, we stepped off the train, then went looking for the room I had booked earlier (My Airbnb coupon code if you want $25 off your first booking:  www.airbnb.com/c/kevink151 ) .

Step 5:  Aguas Calientes to the top/entrance of Machu Picchu.
Aguas Calientes is a small village/city in the valley of the mountains.  Machu Piccu is up in the mountains above Aguas Calientes.  There are two ways to get to Machu Picchu.  Hike straight up the mountain on a steep trail, (We did it in 45ish minutes) (an older gentleman told us it took him over 2 hours, it's steep..)  The other option is to buy a bus ticket to ride a bus that takes you to the top of Machu Picchu (takes 30 minutes, costs $12 USD one way.)  Extra Advice: If you do any type of hiking at Machu Picchu, wear pants.  My friend Andy and I wore shorts our first day at Machu Picchu and we didn't really notice anything as it happened, but later we realized we had a bunch of bug bites on our lower legs/ankles.  Later that night and the nights following, these bug bites were insanely itchy, the worst either of us have ever had.  What's worse is they didn't go away for like 2 to 3 weeks after the trip.  The next day it was super hot doing our hiking at Huayna Picchu with pants on, but definitely worth it to avoid any more bites. 

Step 6:  Machu Picchu or Huayna Picchu 
We didn't buy the bus tickets, deciding hiking would be more fun.  It was a hard and really, really sweaty hike, but it felt like we earned Machu Picchu more than if we had just taken the bus.  
You need to buy your ticekts ahead of time for Machu Picchu.  Or if you're like us and have nothing prepared, just show up in Aguas Calientes, go to the local ticket shop, and hope they have entrance tickets available... 
Huayna Picchu is different than Machu Picchu and requires a different type of entrance ticket (costs about 15 or 20 USD more).  Each type of ticket gives you entrance to Machu Picchu, if you have the Huayna Picchu ticket, this gives you entrance to Machu Picchu, which you walk through to get to the entrance of Huayna Picchu.   Hiking around all of Huayna Picchu is much more of an adventure than just Machu Picchu and I would have been really disappointed if I missed out! 

Huayna Picchu is where the adventure's at.  Huayna Picchu was a lot more exciting and much more of a challenge than Machu Picchu.  Machu Picchu you just walk into, and you're there.  It's cool and all, but it's really tourist oriented and just not that exciting.  Charging up the cliffs and narrow, steep stairs at Huayna Picchu however, is a lot more interesting.  It was much more fun to hang out up here and explore.  We just sat up there and enjoyed ourselves for a couple hours, happy to be free of the clusters of tourists way down below, scavenging around the tiers of Macchu like little ants.  After climbing to the top of Huayna, we then went down the back way, and all around the entire mountain and out the other side.  We learned as we hiked that this is the much less common route, there was hardly anyone on it.  This backwoods trail takes you through jungles and some other, lesser known ruins.  Once we were on the very top of Huayna, we could see the river thousands of feet below in the valley that holds Aguas Calientes.  It was quite the journey hiking all the way back down! 

Is it called Wayna Picchu or Huayna Picchu? 
We couldn't figure this out.  All over the place, this mountain was called Wayna and Huayna interchangeably.  The internet seems to call it Huayna Picchu, but even on our entrance tickets and on the sign outside the gate, it's called Wayna Picchu.. Even when we bought these tickets in-person in Aguas Calientes, the guy was confused when we asked about Huayna Picchu.

When is the rainy season at Machu Picchu?
The rainy season in Machu Picchu is during the United State's winter, so roughly around November to March.

What's it like visiting Machu Picchu during 'the rainy season' in November / December.
We were in Aguas Calientes from November 29th through December 2nd.  We went up to Machu Picchu all day on November 30th and then up to Huayna Picchu December 1st.  At Machu Picchu, the first day, it was super hot and verysunny, not a cloud in the sky.  I even got sun burned.  Roughly, it was 75 degrees (F).  The next day we went early up to Huayna Picchu, we hiked as fast as we could to be the first ones up to the top.  Pouring sweat as we hiked up the endless stairs, we eventually made it up.  It was around 70-75 degrees again, clear blue skies with a few wispy clouds here and there.  It was not hazy or foggy at all, and I could see clear to the horizons in every direction.. and Aguas Calientes in the mountain-valley below.  Around noon it got slightly cloudy in the sky above, not much blue left, but still not hazy at all.. Just cloudy.  It didn't rain at all while we were up around Machu Picchu either day.  Down in Aguas Calients the weather was very similar.  One day it was blue skies all day, another day it was cloudy, but still clear, and no rain.  I believe it rained once in the middle of the night for about two hours, and once around dinner for an hour or less (we were there for 4 days)  (The two days I was in Cusco it was cloudy with some blue skies and didn't rain.)

What are the benefits of going to Machu Picchu during the rainy / off season? 
There are soooo many benefits by going to Machu Picchu during the off-season, that after doing it, I wouldn't do it any other way.  All the hotels/hostels are cheaper and have more availability.  The train from Cusco to Aguas Calientes has more openings.  The entrance tickets into Machu Picchu and Huayna Picchu are more easily available as well as the bus up the mountain.  There are just fewer tourists in general, which makes everything better, more open, more available, and cheaper. 

Does Lima airport have free WiFi?
No, it gives you 1/2 an hour free wifi then locks you out completely.  If you want a 24 hour WiFi-pass, it will cost you 10 bucks. 

Where do you get the bus tickets for the buses that take you up the mountain to Machu Picchu from Aguas Calientes? 
When you get to Aguas Calientes, you will see the buses, you can't miss them, they are right in the city center.  Green and tan Mercedes buses that hold 20+ people each.  Next to where these buses line up, there is a little ticket booth/office.  This is where you buy the bus tickets.

How much are hotels in Aguas Calientes.
Aguas Calientes pricing is completely different than anywhere else in Peru, everything in Aguas Calientes is more expensive and similar to prices you might find in America, and even more expensive and more sold-out during the busy season (perks of going in the off-season).  In Cusco, hotels were cheap, and we stayed at Hostels for 9 bucks a night.  In Aguas Calientes, the average for a room with two beds was ~60-95 USD.  They can get a little cheaper for a little less quality, and they can get a lot more expensive if you want something luxurious. 

How much is food in Aguas Calientes vs Cusco.
Same thing, food in Cusco is pretty cheap, but you won't find anything in Aguas Calientes for less than 10 USD per plate when you go out to eat which is pretty much your only option. There are small grocery markets, but it's hard to cook anything without a kitchen and even so, the markets are still pretty expensive.  More than what you would pay for food at a US gas station. 

Is the altitude in Cusco or Aguas Caientes or Machu Picchu hard to deal with? 
I could definitely feel the altitude in Cusco, especially when I was hiking up a bunch of stairs and steep streets with my backpack on.  I didn't sleep quite as well as I normally would, but I was so tired from the flights to get here that it wasn't really a problem.  It's nice being in the Cusco altitude first because then you feel better when you get to Aguas Calientes (which has a lower altitude).  Aguas Calientes feels pretty normal.  The altitude is slightly noticeable, but nothing that will cause any problems.  The place where the altitude came into play the most was Huayna Picchu!  Sprinting up these stairs that are so vertical, they're more like ladders,.. this is where we could really feel the elevation.  It takes a lot of effort, huffing and puffing, and sweat to get to the top of Huayna mountain.. especially after climbing all the way up from the Aguas Calientes valley. 

What is the altitude of Cusco?
11,152 feet

What is the altitude Aguas Calientes?
6,693 feet

What is the altitude of Machu Picchu?
7,972 feet

What is the altitude of Huayna Picchu? 
8,924 feet