A bunch of little things I've noticed regarding getting around in Moscow ...
The underground metro is the usual means on transport inside Moscow, along with trains and buses. You can now purchase a Troika card for all of these, but many of the buses in surrounding towns do not accept it.
If riding the trains, you will receive a little scrap of paper as your ticket - do not throw it away until you have left the station at your destination. Technically buses are supposed to provide you with a ticket upon payment but they almost never do, and it's almost never required.
As with some other things here, cabs are ubiquitous and relatively inexpensive in Moscow, so, if you have some money, it's not a bad way to get around.
If riding a boat on the river, I suggest purchasing your pass waterside, since online sites jack up the price for tourists.
Commute times around Moscow can vary quite a bit, depending on location and road conditions. Some sections of the circular MKAD highway, for example, are are known to be typically slower than others. People in Moscow generally begin work at 9AM and leave at 6PM. If commuting, I'd suggest leaving a bit early in the morning to avoid traffic. What might be a 40-minute drive some days may be a 2-hour drive others. If you can, use the metro.
Many people both young and old use scooters, skateboards, and bikes. Don't be surprised if you see children riding their bikes inside the grocery store. Many of the outdoor walkways are paved with cobblestone bricks, meaning they are a little bumpy.
My personal advice is to be careful in crosswalks - step out clearly while politely but firmly staring at the vehicles as they approach, ready in case a driver does not stop. I was surprised to see that, although some other rules may be disobeyed here, generally everyone uses the crosswalks for their own safety, and maybe because the roads are icy in winter, so not easy to run across. Some of the crosswalks allow diagonal walking, which is pretty cool.
If driving, be careful, because on days with slow traffic, vendors often stand in the middle of the highway selling ice cream and whatnot. Other things to look out for: do not stand in doorways of buses or trains, or they may slam shut on you. it's okay to gently push the person in front of you and say 'razreshitye'. And there are a couple low-hanging bits of concrete at the metro entrances downtown that can give the head a whollup.
I've never driven in Moscow - compared to where I'm from in the US, it looks pretty intense, with cars often trying to push past each other in merging lanes. That said, I've also ridden in vehicles that drove more sedately, so I suppose it's possible. Personally, I was a bit excited to see European car brands that are rare in the US: Renault, Skoda, Peugeot.
I can't speak to parking, but I'd sometimes see cars parked with their wheels on the curb - this practice surprises me, as I imagine it damages tires.