Avoid the neighborhood around the Termini rail and bus station in Rome.
"The" neighborhood around Termini breaks down into a few totally different areas, IMO. There's the neighborhood south of Termini and between the station and Santa Maria Maggiore. I think/assume it's a place where many people pick a hotel because of its location, and the neighborhood as such is sort of nice, I suppose (given that it's smack in the middle of an Italian metropole). We liked the area opposite, north-east of Termini: San Lorenzo. Much less touristy, more oriented at locals (students), more edgy, but also more authentic than the more trodden places. I can't say anything about hotels as I've never stayed in any in Rome. We generally use (& prefer) AirBnB. Resturant-wise, in San Lorenzo the food is decidedly more rustic, better priced and of decent (albeit generally somewhat robust) quality - more so than in the touristy spots. You may or may not like the neighborhood's atmosphere at night. The streets running past Termini draw some shady types. Lots of people living in the streets, too. For the most part, if you don't bother them, they won't bother you.
Museums, shows, restaurants, exploring, taking it easy for us old folks.
If you've never been in Rome, I imagine you'll likely want to visit the highlights. Book a few days extra there - at least.
Book your museum tickets in Rome in advance. This saves you a lot of queueing up, which isn't fun especially on June, as Rome will be very, very crowded and in all likelihood very hot as well. Book at the official website; avoid at all cost 3rd party booking systems that promise lower rates (a lie), quicker access or shorter queues (more lies). Avoid the hawkers in the streets with the same promises (=lies) or who argue you need a guided tour to get around in these museums/sites of interest (you really don't).
Especially if you want to include places like the Villa Borghese and its renowned gallery, be sure to book ASAP and pray that there'll be slots left. The Vatican Museums are another highlight you'll likely want to visit because they give access to the Sistine Chapel, which seems to be one and only reason why about 80% of the visitors of the museum enter the place anyway. The museum itself is definitely worth it; the Chapel is of course beautiful, but it's virtually impossible to enjoy its beauty. It's just too crowded and there'll be guards urging you to move on if you stop for more than 1.3 seconds.
Restaurant-wise, you're good. It's Italy. It's generally difficult to find bad food, although Rome of course is one of the few places where you might actually succeed. At night, go to Trastevere, if only once - yes, everyone does, but it's OK; there's a good reason. Plenty of decent restaurants and the atmosphere really is nice.
If you spend a couple of days in Rome, sooner or later it'll wear you out. Make your way to the street that runs above the botanic gardens (Passeggiata del Gianicolo) and enjoy the view. Or, also a great view, in this case overlooking the Tiber and Trastevere: the Garden of the Oranges (Giardino degli Aranci).
Late in the afternoon, go to the Tiber island, there's an ice cream shop on the road that runs across the small island. Get a cone there; the ice cream is good. Eat it at the quayside, looking in the direction of St. Peter's, while watching sportsy Romans run along the riverside.
Can't help you with Istanbul; I've never been there, but it must be beautiful. It's been ages I've been to Barcelona I vowed to never go back. It's OK, I guess, but not my kind of town and way, way too crowded. Watch your purse/camera/anything of value. Seriously. Barcelona has a reputation to uphold in this regard, and it's the one thing it does with fervor.
Enjoy the trip - it'll be HOT, but no doubt worth it. The Mediterranean is gorgeous.
PS: I hope you'll pass the Strait of Messina during daytime. It's a nice spot, with the volcano towering over Sicily on one side and the hills of Calabria ('the toe of the boot') on the other.