Accommodations other than hotels

Train stations often have luggage storage.
Renting an apartment in Rome is just like renting one aim anywhere else. Keep your wits about you, but I would not considered it particularly unsafe. If you are more comfortable with a hotel, there are plenty. That would solve your luggage issue.

Easy to find a hotel in Rome with a 24 hour desk and luggage storage. But in smaller towns there isn’t always a typical hotel but often apartments or just rooms inside buildings where you are on your own for the most part. I have stayed solo and with other people in all kinds of settings and have never felt unsafe. Check out the listings thoroughly before booking so that you know what you are getting. You might feel more comfortable staying in a typical hotel to have a host for support and information, rather than a room or apartment.

Stacy, while our short-term apartment rental experience has largely been in the U.S. I wouldn't expect your host to meet you early just to drop your bags: expect to do that at the train station and go back to retrieve them at check-in time. So far, we've not been accommodated for a very early check-in (thus would not be accommodated for bag drop) when renting in the U.S. and I'd expect the same abroad.

Hotel bag drops are almost always handled by desk staff so....

Safety? Apartments can be perfectly safe but I wouldn't rent anything that didn't have a lot of reviews (that don't sound written by the same person), and I would run a check on the photos to make sure they don't show up on other sites for unrelated properties. First-timers abroad can also feel a bit more secure with a desk staff to answer questions or otherwise provide assistance if a problem arises.

Booking.com has been my go-to resource for both hotel and apartment searches, and oftentimes reservations.

Last 2 Italy trips, we’ve mostly had apartments. Sometimes the owner met us to check in, give us the keys, show us how to run the washing machine, etc. Sometimes it was an agent who worked for the owner or the rental agency. Either way they were happy to answer questions, give tips, explain where to buy groceries, good restaurants in the area, and so on. So a 24- hour desk wasn’t essential. Make certain to close the door that leads out to the street behind you, coming or going out, even in the best neighborhoods.
It would be nice if that person would let you stash your bags at the apartment in the morning, even if it hasn’t been completely cleaned after the last people have checked out. But you’ll probably need to wait until they’ve check out and vacated the apartment themselves, so it could be 10:00 or 11:00 AM, at the earliest, before you could expect to be able to drop off your luggage at an apartment.

I generally avoid apartments because they tend to be harmful to the local real estate market, as well as disturbing to the immediate neighbors (if there are any left.) My suburban NJ town prohibits short-term rentals for that reason. I write especially as a long-term apartment resident. (Private home today. But think about what was "nice" about where you grew up.)

You can't talk about "empowering financially-stressed locals to monetize unused space" when over 40% of the short term rentals in an area are run by owners with two or more properties on offer. That's professional real estate investing ...

I’ve travelled a lot and have stayed in over 30 rental apartments overseas.
What I do is book a hotel for just my first night, after a long day and night of flying to Italy, France , Denmark, wherever.
That way, I feel safe, I can leave my bag if my room isn’t ready early, and I get a night’s sleep to feel “normal” the next morning.
Then….the next day, I find my vacation apartment, and check in with a clear head and my wits about me so I can take in all the instructions, how to lock the doors, etc.
I find I feel better doing that rather than landing in a strange place and having to deal with a new host in a different language when I’m exhausted,not thinking straight, and vulnerable.
I book the hotel as near to the apartment as I can.
Maybe that would work for you, especially on your first trip to Italy?

Beyond left luggage facilities at train stations, you might check out an online service such as Stasher - they have locations throughout major cities in Europe.

Consider a monastery/convent accommodation. My wife and I love them. www.monasterystays.com

Convents and monasteries are interesting to stay in, but they do have rules, such as curfews which aren’t for everyone.

Stacy, I'll throw this one out for Rome:

https://www.the-beehive.com

This hostel has seen really good reviews for a long time and offers private, ensuite rooms as well as dorm accommodations so take a look at those?

https://www.the-beehive.com/dorms-rooms

You would have access to a communal kitchen/common area not available to the dorms. This hostel is also nicely located just a few blocks from Termini train station, and offers "Aperitivo Evenings" social hours with free pizza and other activities you might find fun.

https://www.the-beehive.com/activities

They do not have a 24/7 desk but have generous check-in hours, and you can tell them when you expect to arrive. See details:
https://book.octorate.com/octobook/site/reservation/index.xhtml?siteKey=b3d6d6b348e42dbf35012be1593657ec

Reviews on Hostelworld, and booking.com (my go-to booking/review site):
https://www.hostelworld.com/st/hotels/p/82786/the-beehive/
https://www.booking.com/hotel/it/the-beehive.html