I assume your 6+ year old Schwinn GTX1 looks pretty similar to this new one in terms of having the same kinds of components and whatnot.
To change the tire(s), you'll want to:
- Release the V-brakes (squeeze the arms together and pull the little curved piece of metal sleeve containing the cable out of the bracket attached to one of the arms) so that you can get your tires past the brake pads.
- Open the quick release lever and unscrew the other end of the axle until it is loose enough on the dropouts, then pull it off the bike (untangling it from the chain if it's the rear wheel).
- If it isn't already, deflate the tire by removing the valve stem cap and pushing on the little metal pin in the middle
- Pull the tire and inner tube off the rim, which should be relatively easy and probably not require tools on your style of bike. If necessary, use a set of bicycle tire levers (or, in a pinch, something like a flathead screwdriver) to lever the tire bead over the hooked edge of the rim.
- Put your new inner tube inside your new tire, and partially inflate it so it isn't flat and flopping around (which would put it at risk of being pinched against the rim). For bonus points, align the logo on the tire with the valve stem on the inner tube.
- Reverse the previous steps to put the tire back on wheel, and then the wheel back on the bike. Make sure the quick release levers are oriented up and a little towards the back so they don't get caught on anything, and don't forget to reconnect the V-brakes or you'll be in for a nasty surprise when you ride it and try to stop.
The tricky part about changing a bike tire is getting the tire on and off the rim. I didn't go into any real detail about that, because (again) your style of tire shouldn't be that difficult, and also because it's the kind of thing that's much more easily shown visually than described. I have faith you can manage it.
As for other maintenance after letting it sit for 6 years, in addition to lubricating the chain I'd also maybe put just a little bit of chain lube or 3-in-1 oil on the jockey wheels and hinges of each derailleur. Your brake and shifter cables might benefit from some kind of very light lube where they run through the cable housings, but probably not.
When you ride, pay attention to see if any of the other moving parts are squeaky or "crunchy" feeling or anything like that. If so, make a new thread about it because servicing stuff that gets greased instead of oiled, like the bottom bracket, is a little more complicated.