Here is a roadmap:
1 -> any
2 -> 5
3 -> 6
4 -> 1 or 5
5 -> 1
6 -> 2
If you start with the tonic or root -- the I chord -- this chord can go anywhere. It can go to the 2 chord, the 3 chord, etc. If you are on the V chord -- the dominant -- there is a strong urge to resolve back to the root. The V has the most tension in a key.
If your next chord is a iii chord, there is a pull to go to the vi. If you are on a vi, there is a pull to go to the ii. Once you have the first and second chord, the rest of the progression is pretty logical for most songs.
This recipe is nice, but if you know your circle, it is even simpler.
There is one common exception: If you move from the I (e.g., C) to the IV chord (ergo F), resolving back to the I would be a very quick chord progression, so you can move to the V (ergo G) to build up more tension which will pull back home to the root. So from the IV, you can swing over to the V and back to the I. This is how the blues works: C-F-C-F-G-C or I-IV-I-IV-V-I
In any other case, you are going to move in a 3-6-2-5-1 progression. If you move all the way to the 3, you will be traveling back home via the 6, the 2, then the 5, before resolving at the 1.