IMO if a freestanding handstand is your goal then you should learn to kick up away frim a wall as soon as possible. Kicking up to a wall gives you a crutch and wont allow you to feel the sweet spot between under kicking and going over. Either use a crash pad or learn to round off straight away and you can start kicking up anywhere. A few tips i found very helpful when starting is to look at the floor, not through. Keep your arms straight and hands slightly wider then shoulder width, this will help with handstand presses as well because you need a wider stance to do them. Keep your feet together and keep your body very rigid, butt tucked. The three points of movement should be a little play in your hips to correct from going under, shoulder for balance and hands to press your fingers for going over and into your palms for going under.
Balance and constant practice is key to holding a handstand but i agree with the OP that strength is going to help a ton and thats where wall walks help. Also wall runs where you walk your body up the wall to a handstand and then alternate your hands touch your shoulder. Of course handstand presses against the wall is key as well. Tons of ab work is also essential, L sit work, leg raises, planche work, planks, etc.