The first thing to do is to hold both crank arms, one in each hand, and try that same side-to-side rocking test. If pushing on one crankarm makes the other one move, you have a bad bottom bracket. If only the arm you are pushing on moves, the problem is probably your crank set.

You can do a lot of bicycle tuning and repairs with just a 5 mm allen key but cranks and bottom brackets don't happen to be one of those jobs. You'll probably need a crank puller and some bottom bracket tools and there's several different kinds. Your new bottom bracket may even require different tools to install than your old bottom bracket took to remove. Cranks and bottom brackets are one bicycle job that I do using a torque wrench. The torque spec for both parts is usually around 30 ft/lbs. that's quite a bit more than I'd use if left to my own judgement. If you don't get them tight enough, you may find your brand new left crank arm falling off and needing a replacement. Torque a square taper crank arm too much and you can make it crack at one of the corners.