Hex keys (aka Allen wrenches) are possibly the most common tools you'll use on a bike made in the last few decades. You'll want a metric set, and thankfully you can often find basic sets under five bucks. Since you mentioned Home Depot, HERE is one option you'd find there. To list some items you'd need hex keys to remove/install/adjust, how 'bout handlebars, stem, brakes, shifters, saddle, newer cranksets, so on, and so on.

You'll also want a set of combination wrenches. HERE is a cheap set, although I'd personally suggest spending a few more dollars for something a bit nicer. You'd typically need wrenches for pedals, the nuts holding your shift & derailleur cables, and all sorts of miscellaneous stuff. (Not just on the bike, but around the house and in your car, too.)

Moving on to bike-specific stuff, tire levers are a no-brainer. We all get flat tires every once in a while. Speaking of flat tires, I wouldn't normally think of a pump as a tool, but figure I'll mention it just in case; a good floor pump is a must. Same for a vulcanizing patch kit like the ubiquitous Rema kits.

Everything else I'd probably consider optional for most people. I've got a feeling I'm overlooking something, but I'm sure others will chime in. EDIT: Some of them type & think faster than I do and have already done so!