You don't need new wheels just because the current ones keep going out of true. You need to find a good wheelbuilder that knows how to properly true and tension a wheel. What you're describing is THE classic symptom of uneven spoke tension. You're at risk of breaking spokes also, until you have that addressed by someone that knows what they're doing. You might find that your current bike shop doesn't even own a tensiometer, or have someone that knows how to use one. Ask them.
That said, wheels can definitely make a difference in the way a bike handles and performs so if the current ones are heavy or limit your tire choices, that's a valid reason to upgrade. But don't buy a new set of wheels because your current bike shop doesn't know what they're doing. The same is true for the rest of it. You don't need a new crankset just because your current bike shop can't properly diagnose why it keeps loosening. Maybe you just need a new bottom bracket.
If cost is no object that's one thing, but it sounds like you have a budget. I'd tend to focus on just getting it working in its current form, and just ride it. Then see what kinds of "upgrades" or modernization really make sense. That's a pretty decent bike and you might find that you like it better than you think you will. If you take it to a decent bike shop and have them do a complete overhaul, that should easily come in under $200 and might be the best way to spend money on that bike, for now.