I recently had a flat tire, so I took off the front wheel it was on and took it to a bike shop to be fixed. While there, the bike guy showed me places on my old tire that were worn out. So I bought a new tire that looked like the worn out one. That was installed on the wheel along with a new tube at the bike shop. When I got home and tried to install it on the bike a couple days later, I wasn't able to fit it betewwn the brake pads on front wheel. After looking at it and looking at the rear tire, I noticed the new tire seemed wider and didn't fit through.. Is there any ways other than getting another tire to do some adjustments to brake pads or whatever to make the tire fit?
It depends on the brakes. V-brakes: you can disconnect the cable (well, the "stiff" part of the housing) from the brake, so that the break pads move to the outside. Once the wheel is built in, don't forget to reconnect the cable!
Cantilever and other (non-road) brakes: either loosen the cable at the adjusting barrel, install wheel, readjust cable tension or let the air out of the tyre, install wheel, pump it up again. I have to do that when installing my cross wheelset in my old Peugeot road bike, not only is there no room between the brake pads but also the clearance at the seat tube is too small (I admit, bike was built for 23mm road tyres, not for 30mm wide cross tyres).
For completeness sake: modern road brakes have a little lever that loosenes the cable quite a bit to allow removal of the wheels.
Do not (mis)-adjust the brake pads. The position of the brake pads is determined by the position of the rim.