Wow, that subject could get big. Everyone has their own favorite ways of setting up throws. I have taught kids judo for quite a while and set up is partially determined by the individual, their body type, and personal preference. Still here are a few things that seem to work well for my kids:
Tani otoshi - this is a great counter throw. I particularly like it as a counter to koshi waza. As your opponent enters for a hip throw, grab them by the back of the gi and drop to tani otoshi. The key is timing.
Osoto gari - My sensei is fond of saying, "all roads lead to Osoto gari." The first road I send my kids to investigate is the one their opponent pulls them down. If you like to set up Seoinage from a push, see the set up of Osoto as being pulled. when you opponent pulls step in driving their weight to the corner and throw osoto to the rear.
Uchi mata - This is perhaps my personal favorite as a throw, ask my training partner. But it isn't great against opponents taller and longer legged than I am. (I am fairly small.) I prefer to execute Uchi mata by setting my opponent up to turn. (Since I no longer compete and train in aikijujutsu, I do this with joint pain.) My kids prefer to use it when their opponent pushes them, they pull with the energy and turn in to throw.
One of my kids has great success with an uchi mata, tai otoshi combination. She normally fights right handed so I will go that way. When her opponent pushes, she pulls them and turns into right side uchi mata. If the throw is not immediately successful, she sets her right foot down inside and pivots on it to throw her opponent in hidari tai otoshi with the same right hand grip. It is very fast and the right hand grip with the left hand throw add a lot of power.