Converting from MTB style levers to STI This is my city bike:
https://www.giant-bicycles.com/en-us/bikes/model/escape.3/18736/76133/#specifications

Currently I use trekking bars and the MTB style shifter/brake set that came with it. If I wanted to convert it to STI brifters as far as I know all I have to swap are the bars, levers and brakes. Is that correct? I'll leave it as a 7 speed and use the tourney brifters, I'm just trying to see if I need to swap any other parts.
Shimano road and mountain shifters use a different cable pull ratio for the front derailer. It can be made to kinda sorta work, but it won't perform right. If you buy any of the currently made Shimano road triple FDs, you will run into the problem that their minimum tooth difference between the middle and large ring spec is 11 or more, whereas you need 10, and also the profile of the cage won't be ideally matched to your chainrings. The best solution is getting an IRD Alpina FD, presuming you definitely want to stay with your current crank. You may also get acceptable performance with an older 7/8s road triple FD such as FD-3303. Older ones have a min difference spec of 10.

Other than that, no issues. Note also that you can use an 8-speed STI if you want, such as Claris.