Having done several Italian holidays in summer I can let you know what works for me.
Rome is hot in summer, very hot, at lots of the historic sites there is very little shade and the sun reflects back from the stone so be prepared for the heat.
For the forums, colosseum, Palatine Hill, it is gritty, dusty and uneven underfoot.
You will walk miles, comfy supportive footwear is the number one priority. Birkenstocks worked well for me, sketchers nearly ruined the holiday as they lacked support. Sturdy sandals or trainers are best depending upon your preference. Have more than one pair so you can rotate.
Good sunglasses are essential. Consider a hat too.
Rome is touristy enough that no one will judge you wearing shorts, cropped trousers or whatever. There will be lots of Americans in shorts to just above the knee with trainers and sports socks, a baseball cap and a t-shirt so don't worry too much about being as stylish as the Italians.
Linen trousers are good. Shorts are ok but not for religious sites, go for longer ones so you don't have bare legs on hot seats. A light floaty sleeveless top feels better than a tight vest.
I find dresses best all round. I look for lightweight fabric, fairly loose fitting, with enough coverage to wear a normal bra. I don't worry about shoulders showing because I don't burn easily and pop a silk scarf or cotton sarong in my bag for any religious places. You might prefer shoulder coverage to protect your shoulders from burning. I wear thin chub rub shorts underneath. I feel more put together in a dress too.
Avoid jumpsuits due to going to the loo.
Take a reusable water bottle and refill it at the fountains, they are all drinking water.
Go for a cross body bag or one that tucks under your arm and one that fastens properly. Pickpockets can be an issue.