OK. In the UK, I lived in Jeans and tee-shirts in the summer, with a jumper in winter and black trousers with a thin smart jumper, and struggled in the summer. You sound a bit like me in terms of clothes.
Sounds like your current wardrobe will be fine for the winter.
So, your looking for summer clothes, and don't wear dresses. So, buy some trousers.
I live in linen mix trousers. Try Next, M&S, Sainsburys, Tesco. This summer I have bought some shorts to take back, mainly because I'd like to dunk my legs in the pool when the kids are in, and can't in trousers!!!! Next and M&S have some roll bottom ones so you could adjust the length and they fit my thunder thighs
Consider tops with sleeves. My shoulders burn, so I cover them with clothing rather than sunscreen
You need a hat. The 'bald" bit in my hair where the parting and fringe split catches the sun, and is hard to cream....
You need a pair (or 2) of sunglasses.
Shoes.... Karimore or Teva walking sandals
, sketchers go walk, and a pair of sandals which won't be coming back with me as they are falling apart. After lurking in far to many sandals you can walk in threads, I'm going to try these after testing in the park, I'm concerned about how much sand they will hold. We will see.
Do Gap do some chinos (and maybe cut off chinos) in a style similar to your jeans???
Go for colours you think are too light. The difference between navy and mid blue trousers in a hot day in bright sunlight is amazing.
Ideally avoid black and navy blatently ignoring the fact Ive just bought navy trousers and shorts in desperation as white is too far out of my comfort zone
I have no style and shouldn't be posting on S&B other than to ask for help, but I do live in an absurdly hot place!
Good luch shopping