I like Gabrovo, but have only been there for visits, and to rent a place. I haven't spent much time in VT at all, so I cannot really compare the two, but they are close enough that you can spend time in both of them while living in VT and see which you prefer.
If you are staying at a hotel, they generally take your passport info and register you with the police. Then, once you move into a place of your own have 5 days to register short-term with the police. You take your passport and rental contract to the police station. Then, you have (I believe) three months to start the permanent residency process. You will have to renew your permit every year, for the first 3-5 years of living here. I would not rush into the permanent registration and take those three months to find a friend to go with you to translate and help you get your documents together.
For EU residents it is rather easy. I think you only need your passport and a rental contract in your name. (Although I am not 100% certain). You will also need a copy of your son's birth certificate, translated into Bulgarian. For non-eu residents you need a bank receipt showing you have 6 or 12 months of the minimum wage (345) for each family member, your visa, a copy of your criminal background check, birth certificates, marriage certificates etc. and the fee is 500 lv a year, plus 45 for your residence card and a 10lv processing fee. (For eu residents I believe most of the taxes are waived).
Finding a house to rent in Bulgaria can be a lot more difficult than finding an apartment. We were renting a house in Varna and are only now switching to an apartment- we have a 9 month old, and I also wanted the house with a yard for him. But something to keep in mind is that it is very common for Bulgarian families to raise their children in apartments... so there will probably be plenty of small playgrounds and parks near any apartment building if you end up in one. Houses are rarely given for rent (mostly just for sale) but it is possible to find one. However, apartment living has many benefits in Buglaria, including cheaper heating costs in the winter, more playmates for your child to meet etc. With the money you save renting an apartment you could go to a villa in the mountains or by the sea on most weekends to give your son plenty of outdoor time.
Just some thoughts. Good luck on your move.