Those are friction shifters , mounted on the stem they got the name suicide shifters as shifting in that location is a PITA. Higher quality bikes had those on the on the down tube. You can get shorter cables and move them to the down tube. If you want thumb shifters that's a whole different SIS set up and most likely you would need to change the rear derailleur to a SIS one. Depends on the system.
Never Give Up!!!
If you are going to stay with friction shifting (shifters do not 'click' into place) then the change is simple. You need to get shifters that have a friction mode and you will probably need new cables and housing as the path up to the bar shifters will be a bit longer than the current path. Setting them up would be the same process as changing the shifter cable so if you look for instructions on that it should cover it.
If you google "thumb shifter friction" you'll see choices ranging from $6 - $100. Pretty much anything will work so I wouldn't waste a lot of money on expensive shifters. (It is worth it to get good cables and housing though.)
If you want to go to indexed shifting, you would need shifter, derailleur, and gear cluster that all match. Not necessarily hard, but we'd need more info on what your bike has currently to give any specific advice.
Note that going to straight bars you will also need new brake levers and new brake cables as well (but the brake housing may be fine.)