Definitely the Brad nails. The finish nails will leave larger holes (16 gauge vs 18 gauge usually) and will be more likely to split the wood. Based on the size of your shoe, i dont think you could use pin nails (18-23 gauge).
The Big Box home improvement store always sell kits with finish, brad, and pin nailers for pretty cheap. Some have 2 guns, some 3, and some come with other stuff. I've always found that the best way to go.
Just did a quick check for some kits. Home depot 3 piece Kit for $99. I have had good luck with the Husky Brand from HD. Lowe's had some kits but they were name brand and a lot more expensive. Could also check out Harbor Freight, their stuff is definitely cheaper quality, but maybe that is ok for your situation and usage.
There are a whole lot more types of boards than that! Even at the big box stores.
Furring strips come in limited sizes and are not meant to be structural or seen.
Pressure treated boards have a chemical applied to them under pressure that helps presserve the wood against moisture and insects. It's generally used outdoors or in direct contact with concrete otherwise it's not recommended to use pressure treated lumber. The appearance, quality and sizes match other framing lumber used for studs, juists, etc. It would be cheaper to use non-pressure treated lumber.
The better wood to use are appearance grade boards. They'll either indicate they are "appearance" grade or described as S4S (surfaced 4 sides). These will have a smooth surface on all 4 sides. They come in a variety of different sizes and species. The cheapest price is for something the box stores call "whitewood" which is whatever light colored wood they can get the cheapest at that time. Pine, spruce, fir, etc. I think Home Depot might be calling them "common boards" now.
The select grade boards are also appearance grade S4S boards but usually of better quality with fewer knots, hence the higher price. Not necessary for a garage cabinets.
That design you linked to looks like it would be expensive to build. If you want to save money and don't care about appearance you can save money by using a different design that uses 2x4's and 3/4" plywood for the shelves.
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I would go with the wood. It will paint & caulk better. Some of the PVC stuff doesn't cut well on a chop box, has a tendency to break and may crack with larger nails. PVC just looks cheap when its up, you can tell its plastic.
If the composite is MDF, I would buy that over wood. MDF is cheaper, and usualy straighter pieces. It also paints nicely. The only time I might choose wood over MDF is if I were staining the molding.