I can pretty much echo your entire post. A 3 and 1 year old, 3 year old loves kindergarten (and I love that it costs about 10% of minimum wage to send him there).
We are thinking about enrolling our son in some extra curricular programs next year. His school offers basketball, football, dance (traditional, ballet, and ballroom), singing, majorettes (for boys and girls), yoga and a few others for a small fee. Of course, the instruction is not as good as it is in the private clubs, so we might pay a bit more and opt for that instead.
Private offerings include everything I listed above as well as various martial arts, swimming, languages, art, cycling, orienteering, scouts, etc. All at different ages and skill levels.
In our city (Gabrovo) music lessons can start around 4 years old for piano. I haven't looked into many other options.
There is plenty of hourly daycare (centers and individuals) at between 3-6 lev an hour, per child.
The extended family jumping in to help out here is amazing. Even though my mother-in-law lives four hours away, we go visit once a month and she is always so helpful with the kids. In fact, she took my 3 year old for the week to give me a break this month. This is something that isn't so common in the US.
Things my son likes doing here? Going to the beach in the summer. Going camping. We take him with us to a scout camp we volunteer at every summer (for about 10 days). Ice skating- Varna has a year-round indoor rink, most cities have outdoor rinks in the winter. He likes going on bike rides (on the child seat of our bike). He loves working in the garden with his grandparents and cooking with me.
Bulgarians tend to be strict about gender roles in children, which is a little difficult for us as both me and my husband are quite progressive. Ex. my son likes painting his nails. I let him wear black nail polish to yasla one day and the parents freaked out about how we were ruining him and next he would be asking for a pony tail. (As a side note, his father has a pony tail, so not seeing that as a big deal).
Not sure how health care is for non-Bulgarian citizens, but I have liked the health care available for my kids. Monthly checkups the first year, every three months the second, and then yearly. Then, whenever they get sick, just drop in to see their pediatrician, no appointment necessary. (All free, of course). They have a different vaccine schedule than in the US, and the anti-vax movement is pretty strong here.
Here people seem more free with their kids. People will leave the stroller outside a store or outside the kindergarten in good weather, not afraid that someone will kidnap their kid. Young kids (7,8) walk to school on their own, and I always hear kids playing outside late into the evening during the summer.