It takes a bit to get going in the game, but you can make do with Normals and such if you choose your opponents very carefully.
For quick boosts to ATK, you might find someone that matches your producer type that gains ATK quickly at low levels (Kyoko Igarashi, for Cute types), and boost them until you see diminishing returns. Mostly though, you can put the highest Cost idols you have in your front group and you'll be as good as you can be (if you level them, of course).
The more long term solution is of course to pick a few idols that offer high Cost and good ATK/Cost and max out two copies of each, then upgrade them and max the plus version, but that can take a bit of time. Just be sure to build up your DEF first. Put the most intimidating idol you have as your leader, and put a lot of points into DEF. People will start leaving you alone.