I'd start by seeing if I could isolate the problem to the shifter, the derailleur, or the cable run.
Does your bike have a bare cable run along the frame's down tube or top tube? Shift into the lowest chainring, then pull the cable by hand, away from the frame, like you're drawing a bow and arrow. Does the derailleur move? Does it move back freely when you release the cable? (If you can remove the chain, it may be helpful to do so in order to observe the derailleur over its whole range of motion.) If the derailleur moves easily enough when you pull the cable and returns on its own, then it's probably OK. The lower cable run is probably OK, too, which means we can move on and look at the other half of the equation.
To troubleshoot shifter operation, I'd want to know a little bit more about it. You mentioned it's a Tourney shifter, but that unfortunately doesn't narrow it down too much. There have been all sorts of Tourney-branded shifters. Regardless of what kind it is, you can try isolating the shifter and cable. One way to do that is to disconnect the cable from the front derailleur. Then, while pulling the free end of the cable with one hand, try to shift with the other. Between your two hands, you should be able to feel if it requires excess force to shift.
One thing you don't want to do is fiddle with the limit screws on the derailleur willy-nilly. Adjusting them wrong can cause your chain to derail completely, dumping the chain off the chainrings. A derailed chain can jam and cause other damage, particularly if it falls off toward the inside. If your front shifter is indexed, maladjustment can cause other shifting issues. For example, if you set the high limit too far inboard, you can make it excessively difficult or even impossible to shift into the big ring. (Come to think of it, kinda sounds like what you're describing, but you'd need to troubleshoot some to know for certain.)