Since I broke up with my ex a few months ago, I’ve been feeling lonely, so I've begaun to participate oin new groups and activities to meet new people.

"I began" vs "I've begun" - "I've begun" emphasises the ongoing nature more, i.e. that you started participating in new groups and you still are participating in new groups

I would say that the most “productive” one haves been the monthly book club.

"one" is singular, so "have" should be "has" instead

I’ve met super funny and interesting people, and also have been interacting with faces I already know.

OnDuring the three club sessions I’ve attended, I(I've felt at home / they’ve felt like home), as if I’ve known these people sincefor years.

The clubs felt like home, as both the clubs and home are places. You feel "at home", but places feel "like home".

"since" is used with a specific point in time, but years is a measurement of time rather than a point in time, so can't be used with "since".

I also would like to join to a dance class, since that is something I’ve always wanted to learn.

Since I brokebreaking up with my ex a few months ago, I’ve been feeling lonely, so I began to participate on new groups and activities to meet new people.

I would say that the most “productive” onegroup haves been thea monthly book club.

On the three club sessions I’ve attended, I’ve felt like home, as if I’ve known these people sincefor years.

I also would like to join to a dance class, since is something I’ve always wanted to learna long-term goal/desire.

Meeting nNew pPeople

Since I broke up with my ex a few months ago, I’ve been feeling lonely, so I began to participate oin new groups and activities to meet new people.

I would say that the most “productive” one haves been the monthly book club (so far).

I’ve met super funny and interesting people, and. Additionally, I have been interacting with faces I already knowmiliar faces.

On the three club sessions I’ve attended, I’ve felt like home, as if I’ve known these people sincefor many years.

I also would like to join to a dance class, since it is something I’ve always wanted to learn.

You don't need a comma before since.

A friend wants to join too, but she always says that she’s busy, so. Therefore, I guess I’m going to give it a try all by myself.

Meeting nNew pPeople

Since this is a post, the original is correct. This correction is just showing how proper title capitalization would look.

Since I broke up with my ex a few months ago, I’ve been feeling lonely, so I began to participate oin new groups and activities to meet new people.

I would say that the most “productive” one haves been the monthly book club.

I’ve met super funny and interesting people, and have been interacting with faces I already know.

Onf the three club sessions I’ve attended, I’ve felt likeat home, as if I’ve known these people sincefor years.

Another option instead of "I've felt at home": "they've all felt like home".

I also would like to join to a dance class, since ithat's something I’ve always wanted to learn.