We did go independently, but also took advantage of a tour, so we didn't feel completely on our own when it came to the basics (simple things like currency, geography and culture were useful from a local's perspective)
The prices of being out and about were quite unusual. Kotor, for example, is a prime destination for tourists, including a few cruise ships, so some of the bars/cafes are more expensive than you'd expect for the region, but still not "expensive". Everything felt like it was expensive, but then we you worked it out in GBP, it was OK.
Budva was interesting. We ate in a couple of places, the food and service was great, prime location and the food was very reasonable. Montenegrin wine is lovely and when I was handed a wine list in both places I noted a huge divide in the prices on the list. The top half had wines around £5 to £15 and the bottom half had wines from £300-3000.
I thought it was just a regular/fine wine split, but the waiter explained to me (pointing at the huge Russian owned super yachts berthed outside, that his wine just wasn't good enough for some of his newer customers. So these days, it's actually possible to do Montenegro on a modest or huge budget in the same places.
Kotor would be a lovely place to take the kids (Safe, clean, friendly), but i'm not sure how much there would be for them to do. If they'd be happy spending long days playing around by the water then it will be fine, but if they'll more likely to want to visit attractions then I've not idea what's on offer.