About horseback riding near Rome:

Lago di Vico is within 45 miles from Rome, and there is a road that goes around the lake
that is (for half of it) closed to cars.
I've been swimming and had long bike rides around there as a boy, beautiful places!
link: <a href="https://www.ilmioangeloranch.net/" style="background-color: rgb(255, 255, 255);">https://www.ilmioangeloranch.net/</a> ; (I had not direct experience with them)

Other places to check for horseback riding near Rome:
Lago di Martignano and Lago di Bracciano,
Tarquinia (beautiful medieval city near the sea)
Lago di Bolsena, not really near,but could be interesting to check. I usually go there for some long bike ride.

After the first week of your stay in Rome, please take a 60 minutes train to Orvieto,
it is not far from Bolsena lake, and can be awesome (I usually love to spend my time studying or
programming, not really interested in art,  but Orvieto is -- IMHO -- just a terrific place,
you can just walk around the town).

A very pleasant place in the mountains of Abruzzo where horseback riding is possible is Pescasseroli,
from there you can go rowing on Scanno lake. 

<a href="https://www.google.it/search?hl=en&authuser=0&tbm=isch&source=hp&biw=1756&bih=874&ei=-1m-XIb1MMGDsAeCja6QCg&q=orvieto+beautiful+images+and+bolsena&oq=orvieto+beautiful+images+and+bolsena&gs_l=img.3...1576.1576..2080...0.0..2.67.67.1......2....2j1..gws-wiz-img.....0.PtlAyM6lOPg">https://www.google.it/search?hl=en&authuser=0&tbm=isch&source=hp&biw=1756&bih=874&ei=-1m-XIb1MMGDsAeCja6QCg&q=orvieto+beautiful+images+and+bolsena&oq=orvieto+beautiful+images+and+bolsena&gs_l=img.3...1576.1576..2080...0.0..2.67.67.1......2....2j1..gws-wiz-img.....0.PtlAyM6lOPg</a>;

@Julia
Glad to be of some help.
Sono contento di essere di aiuto.
(We actually say: sono contento di esserti stato di aiuto)
esserti: essere+ti = essere a te
Thanks to you!

When in Rome, please, please,
do as you will do in your Country!
That will be much more interesting for us!
Some of us are genuinely eager to learn.

You talked about something related to food:

have a look at these guys: gnammo.com

especially here:  <a href="https://gnammo.com/Janice%20&%20Francesca" style="background-color: rgb(255, 255, 255);">https://gnammo.com/Janice%20&%20Francesca</a>;

and at these other guys: <a href="https://www.eatwith.com/events/23980?date=2019-05-09" style="background-color: rgb(255, 255, 255);">https://www.eatwith.com/events/23980?date=2019-05-09</a>;
(btw I'm also in that picture)

I'm talking from my direct experience! It's fun, you will end up speaking in English and Italian.
You will eat very good Italian food. Sometimes you might even find a 'cooking class'.
I met several people from England, Korea, Australia, USA, Canada, and (obviously)  from Rome in those events.

Wish you have a pleasant stay in Italy!