Re: all spokes are loss and wheel wobbles side to side
Cheaper wheels are hit and miss with their initial tensioning. Too loose and the nipples on the ends of the spokes will loosen over time. Too tight and the spokes will break at the bend where it's attached to the hub. A little of both and you get a little of both. Usually the loose spokes loosen up until the tight spokes are carrying too much of the load and start breaking. I have a few sets of anal retentively tensioned, hand built wheels that I beat the "snot" out of and with one exception, they've never once gone out of true.
My local bike shop will true a wheel for $5 and retension it for $12. However, if the rim is already damaged, they will be unable to evenly tension the spokes and no amount of work will keep the wheel true for an extended period. If that's the case, decent cheap wheels can be had for about $30 front and $45 rear and the shop will likely check the tensioning and fix it if it's wrong for free.