If it's the feel of skin tight you don't like, technical fibres are great nowadays, you won't feel them, although you might not get them at supermarket prices. About 1% wear loose fitting clothes, and it looks odd, on all shapes and sizes, honestly. Don't do it.
Lots of ppl will be on to recommend Sports Direct, Matalan and Decathlon, all of which have their place, but if you've joined a swanky gym and as you asked to be indulged, then go for Sweaty Betty, Gap, Nike, Adidas, Under Armour. There are more boutique-y places, but that's a start. Sweaty Betty has a sale on now, Gap have constant discount codes if you sign up.
As a starter, go for black 3/4 length leggings with a bright colour vest and contrasting sports bra. Doesn't matter if you can see it underneath.
Again, as you asked, Nike or Adidas trainers will have more swank appeal, you can pick up a bright, decent pair for around £50.
Obviously brand names don't matter (although with sports wear, the clothes you wear will actually help you stay cool and comfortable, and you'll perform better). I'd recommend you go for a coffee and watch the regulars come and go for a bit if you're really bothered.