OK so you are using a stock width rim, this isn't one of those huge fat, rear tire bikes. A 190 or 200 is probably going to be a tiny bit fatter than a 180. As I understand it, the first number refers to the distance from the edge of the tread measured around the belly of the tire to the opposite edge--just like you used a tape measure to get the distance from the RH edge to the left. 190s and 200s are always going to be a taller tire than a 180 but they are probably going to be a few mm wider too. I'd show a pic but my briefcase is so full it takes hours for it to open with the low internet speed I have.
I have replaced tires 4 or 5 times myself. If you replace your own car tires, I imagine a bike tire would be easier. It takes me hours to do one bike tire and it is quite physically demanding. A lot of people say they can change a tire in 20 minutes with tire irons but I'd like to see what their rims look like inside. I have seen a video one member (Nightmare, I think) did where he changes a tire perfectly in just a few minutes but he must have a great deal of tire changing experience. If you do not, I would suggest paying whatever a good shop charges to change tires rather than invest in a stand, tire irons and other equipment.
As far as removing and installing a wheel, that is easy but you will need a good stand that will cost ~90-$250. I can direct you to threads about stands and wheel removal if you like. If you trust the shop enough to change tires, you probably can trust them to install your wheel. I have never brought either of my bikes to a shop but I did bring my wheels in once. I saw the guy throw the wheel on the ground right in front of me and that was the last time I ever did it. After I removed my first tire, I saw how they scraped the inside of the rim and also the outside edges. The beads were in good shape and that's what really matters. If a shop changes your tires, you can expect at least a few nicks. Then again, I marred my rims a tiny bit the first couple times I used tire irons too.
'08 MIDNIGHT SAPPHIRE BLUE Now Deceased