I recently purchased a 2008 Jeep Patriot. The model that I purchased came with 16" tires and wheels, but I wanted to upgrade to 18" tires. I am having a problem finding out what size will work the best for my vehicle. I also want to make sure that the TPMS will work with the new tires. Does anyone have any suggestions?  

As long as you stay with a similar diameter tire you shouldn't need to mess with speedo or TPMS. a much larger tire would require extensive suspension mods before you'd need to think about displaying accurate speed. Also, the Patriot is very similar to the Compass, which came factory equipped with 18-inch wheels, so fitting them on yours shouldn't be too difficult.

I believe that the 16-inch factory wheels have a 6.5-inch width and 5.5-inch back spacing and a 67.1mm hub diameter. they use a 5 on 4.5 bolt pattern like many other Jeep models
 
18-inch wheels with P215/55R18 tires would probably do the trick. This was a factory size for the Compass. You will need to determine the proper wheel offset and back spacing but most quality tire/wheel shops should be able to help you determine this. If they can't try a different shop.