Sorry - just noticed you say you don't have a car. Hrrmmmm - three weeks on holiday without a car, and doing it on the cheap
There may be some sort of bus pass that you can get that allows you multiple trips in that time - you would then need to pick the locations based on that pass, and where you can get to good stuff by walking.
Walking, in itself, may be a good option if you are that way inclined - there's a good (long) track that takes you from below Adelaide right up through the Flinders. Maybe do portions of that?
Spend a few days having a good look around Adelaide - there's lot's on offer, and the public transport system is (believe it or not) very good.
If you can get up to the Flinders, there's a bit of scope for walking (Wilpena Pound itself can
only be explored on foot).
It's hard to travel cheaply without a car - not to say it can't be done, but you're not going to be able to see as much as you could by having your own transport (or paying through the nose for daily tours).