IEMs: Bowers & Wilkins C5 S2s
- arguably the best sounding IEMs I have ever used for its price range which seems to coincide with your own price range. The secure grip also keeps the headphones in your ears a lot if that is a concern for you and its easily adjustable.
Over-Ear: Audio Technica M50
- I usually prefer not to use over-ears with my phone just because of the inconvenience of carrying around but I really like these headphones. Again looking at your price range and these are the best bang for your buck IMO
I have tried and own IEMs/headphones much fancier than these but if I was considering a budget, these are the ones I would get.
I will agree with others on the Jaybirds Bluebuds X and X2, those are some great workout headphones.