When a rim merely wobbles right and left between the brake pads as you spin it, that is called being "out of true". If it can still spin - isn't out of true so badly that it bind up against the pads - then the rim can almost surely be re-used. If it is more out of true, then could be iffy. If the rim has a visible dent or bend or damage, then it probably should be replaced, unless you want to try hammering and bending it back straight, a tricky thing to do.

If spokes are broken, a rim will go out of true. If spokes are loose, they will break.

When you brought the bike into the shop, they should have replaced any broken spokes and tightened the spokes to bring the wheel back into true and eliminate the looseness in the spokes. In the US, that costs around $30 to $50. I cannot tell if they did that, or offered to do it.

Since you want to learn bike maintenance and repair, and are I assume serious about it, you might want to try your hand at rebuilding one of the untrue wheels. At worst, you can bring that wheel or the other untrue wheel to a bike shop and have them fix it, if your own efforts fail.

You will need a spoke wrench ($5), a flathead screwdriver sized for the spoke nipples, some grease (a small tub or tube will be more than enough), some light oil. If one of the broken spokes is on the drive (right) side of the rear wheel, you will need the appropriate tool to remove the freewheel, and a largish wrench to turn it (the freewheel blocks removal of the spokes on that side). The tension meter mentioned is useful, maybe not mandatory but it will make things a lot easier. You will need to remove the broken spokes, measure their length, and get replacements of the same length, gauge (thickness), and type (straight or butted). Spokes are not the same length from front to back or from drive to non-drive side, so measure each broken spoke, dont assume measuring one spoke works for all the spokes. You will also want several new spoke nipples as some of the existing ones will be rounded off or otherwise hard to work with.

And you'll have to read up on how to replace a spoke and true a wheel. I don't have a link handy. If you go to Sheldon Brown's website and start poking around, you should find something. That website is one of the best ways to learn about bike repair.