Previously I used Nexus Mod Manager and didn't have any problems.

Now with Skyrim on Steam (And using SKSE) and opening the game from the SKSE shortcut (following SKSE directions) instead of from in Steam...where do I put mods to get them into the game?

The same place as before, steam/steamapps/common/skyrim/data. Mods from Steam Workshop install automatically But I don't know why you can't just keep using NMM (or Mod Organizer, I prefer this one).