Concrete – use landscaping lava rock as a concrete slab base

Consider renting a rear tine roto-tiller. They are very easy to use and very effective in breaking up hardened/packed soils. I would advise tilling to a depth of 3 to 4 inches. Buy a landscape rake. This is an aluminum rake that is about 3 foot wide with strong rigid tines for sifting out stones etc, and a scraper edge on the top side that is used to level and move soil. After you have loosened up your soil, you can add amendments, starter fertilizer etc if wanted, level out the area with your landscape rake, reseed, and roll it flat (optional) if you wish. Midwest Rake 10030 Aluminum 30 Inch Landscape Rake apx $40

You can install an interior weeping system similar to what you would normally do on the outside, but without the waterproofing.

Start by jackhammering out about two feet of floor around the edges of your garage, as close to the walls as you can get. Dig down until you reach dirt, and if that's not at least two feet down, keep going. Now, put down a couple inches of 3/4" crushed gravel, and then install weeping tile (it's not tile anymore; it's 4-6" corrugated plastic drain pipe with a nylon mesh sock over it). Make sure the sock is continuous across any joins you make in the weeping tile for corners etc. This is a good time to direct those drain holes down into this trench, where the water they drain will flow into the weeping tile. You'll also need a sump, or a connection to a downhill storm sewer; direct the weeping tile to this sump pit or the storm drain. While you're doing all this a center drain can't hurt too much; you can tie it into the weeper wherever it's convenient, just make sure the garage floor and the drain flange are level, or that the garage floor slopes slightly to the drain.

Now, you can install waterproof sheathing on the interior walls contacting the concrete. Normally this product is designed to go outside the foundation walls, but in cases where that's infeasible it can work this way too. The idea is to trap moisture that weeps through the wall behind the sheathing, where it will then be directed down the wall and under the slab to the weeper. So, you'll install the sheathing, which should have an air space between wall and most of the layer, and make sure the end is down in the trench by the weeper.

Backfill the rest of the trench with 3/4" crushed gravel up to the slab footing, then patch the slab with new concrete. Your garage should stay much drier.