Incredible views over the Bay of Kotor
There are no cars allowed inside the Old Town of Kotor, so you will have to park outside of it and walk in. The Old Town of Kotor is like a triangle pointing to the left, and the trail for this hike can be found up near the top corner of that triangle (the spot I've put on this map for the trail is exactly where to go). You'll see an archway that you can walk through and then the cobblestones will start going uphill a bit. After about 50 feet you'll see a person on the left collecting entrance fees (it was 3 euros when we were there but could change; cash only). Pay him/her and then get on your way climbing!
The climb up involves a ton of stairs (my reading told me 1,355) or you can choose to walk up the angled cobblestone next to them (most of the way up you have the option). There's nothing technical about this hike but you probably want to be in okay shape to tackle all the stairs which is why I rated it beginner/intermediate. The stairs and the cobblestone are all very slippery from years of foot traffic (even more so if it rains), so I chose to stick to the stairs to have less of a chance of falling (as I'm prone to do). However the stairs are narrow and at some points people will want to come down on the stairs while you're going up so you'll be forced to step off onto the cobble stone. Just wear good shoes and be aware of how slippery it is.
About halfway up, you'll come to the Church of Our Lady of Remedy. There were people selling snacks and water and souvenirs here, and there are nice views. You could stop and turn around here, but the views do get better.
When the path seems to split, keep to the right to get to the Fortress. The views from here are spectacular, and there's lots of spots to climb up and take pictures from. Be careful, as it's an old fortress and doesn't have guard rails or anything.
When you're done admiring the view, follow the same route to get back down. Enjoy!