There isn't one right answer to this sort of question, just a bunch of no-doubt-varying suggestions. Here are mine.
I don't think you need to worry about hot weather at that time of year, even in Rome. I've had coolish, wet weather in Venice and Florence even in late May and early September, so preparing for that would be my priority. I do laundry in my room periodically and would take approximately this (which includes what I would wear on the plane).
- 3 pairs of slacks. Mix-and-match with tops to maximum degree possible; take what you already own. One dress would replace one pair of slacks and one top.
- approximately 5 tops, probably all long-sleeved, maybe 2 turtlenecks.
- 1 pair longjohn bottoms (Walmart has a cheap line of Cuddl Duds). Some might substitute tights.
- 1 longjohn top only if you are cold-natured and the last-minute weather report doesn't look good. You can't easily take this off if you're too hot indoors.
- 5 changes of underwear/socks.
- 2 or 3 bras.
- 1 warm layer (could be merino wool sweater, fleece jacket, scrunchable/packable down jacket--or perhaps just a vest, wool jacket, etc.--there should be something in your closet that would be fine). Try for something that is easy to carry around when you don't need it for warmth. I have a boiled wool jacket, but it's pretty bulky.
- 1 waterproof or at least water-repellant layer that will fit over the warm layer. This will also serve as a windbreaker. If the weather report is really grim (cold and much rain expected), I'd consider a full-length raincoat or quality poncho, but those are a royal pain to drag around when you don't need to wear them.
- hat/headgear, ideally at least water-repellant. Consider that it may be windy, too, so umbrellas may not be too helpful, and unsecured hats may blow off your head.
- gloves if last-minute weather forecast suggests temps will be low enough you'll want them.
- scarf if you are so inclined.
- supportive, broken-in walking shoes with grippy soles that will be safe on wet cobblestones. I wear nothing but lace-ups, but some people manage with other styles. Waterproof or not is something worth thinking about. Cold wet feet are no fun, but it may not be that cold, or that wet. Breathable waterproof shoes are expensive.
- other shoes if you must (I take only one pair for summer-long trips, but I don't dress up). If your shoes are not waterproof and are not the type to dry quickly, you probably should have some sort of second pair.
The shoes are critical. If you're cold, you can buy a sweater in Italy. Finding a good pair of shoes is a much more complex (and probably costly) process. If you opt for style over comfort when you pack, you will regret it. You will be walking and standing on very hard surfaces, possibly for 8 hours on some days. Sore feet and lower-back pain are a real risk if you don't wear good shoes that are right for your feet. There is no perfect brand/model of shoe that is right for everyone. You may already have shoes that will be fine if you do a lot of walking at home.
To keep the weight of your luggage down, be ruthless about the toiletries and cosmetics you pack. Decant things into bottles that are just large enough for the length of your trip. Your hotels will almost certainly provide at least hand soap and shampoo. They likely will not provide washcloths, so you'll each need one of those. I take something colorful so it doesn't get mixed in with the hotel's lines (which will probably be white).
There are other handy-to-have things that we can suggest to you, but I'm focusing here on the things we all take, to one degree or another.