Welcome to TE!
If you've got platform pedals, wear whatever shoes you like. Cycling shoes sans-cleat are a little stiffer, but you don't need them, unless you can see yourself wanting clipless pedals one day.
As for your other question: I don't have enough details about where it is, but perhaps try lowering your seatpost a little. If I've got mine set too high, the muscle near my hip joint will be horribly sore for the next few days. If you're a relatively casual rider, I'm not sure compressive recovery gear does that much.



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