What program should I use for item sets, now that they've been removed from the current client?

None. That would be cheating

What I do is look up my match on lolnexus and it shows me what higher rank players have recently built on the champ, then I make a build from there depending on the situation I am in.

I would not suggest downloading a 3rd party program for LoL because that can get you banned quick.

EDIT: It shows you stuff for the champ you picked, you don't see it until champ select.

You can still make them in and access legacy client and still access them during game while using the new one as far as I know. I would log into legacy client, make what you need and save those. Not everything has been switched over to the new client yet so maybe it's coming, just slowly.