So, the absolute first thing that I would spend money on is a pair of rings. Specifically, if you can get your hand on two rings with the Wounding property (with +2-4 Damage on it) as soon as you reach level 6 (which should be the minimum to equip them), you'll notice a dramatic increase in character power. These two rings will cost somewhere between 3500 and 4500 gold, depending on the exact properties (a Scouting Ring of Wounding, with +Magic Find as well is a particularly great grab). Of course, the vendors inventory is random. Fortunately for you however, it refreshes every time you exit the game and create a new one. Once you've rescued Deckard Cain from the cathedral (which is the prerequisite to access the vendor in question), you should log in and out a few times while checking the vendor. Once you see a ring that appeals to you, snag two of them, and you're good to go.
After that, The Blacksmith is probably your next priority. Specifically, you'll want to make sure that you always have him trained up to produce gear either at or just above your current level. That means you'll want to have pumped 8,000 gold into his training by level 10, another 20,000 by level 20, and so forth. You can keep track of all the numbers, as well as the recipes available at each tier over on Blizzards site.
Assuming you're keeping the blacksmith current, at this point, any additional gold can be spent as you please on upgrading your equipment. Whether you do that through crafting, the auction house, or browsing the various shops of Sanctuary, is your call. There are advantages to all of them, and at this point, the game opens up enough that one-size fits all answers are no longer easy to provide. Just remember to keep your artisans (the Blacksmith, and eventually, the Jeweler) up to date, and you're good to go.
And whatever you do, don't worry about the Stash unless you're at the point where you have more gold than you have uses for it and the convenience is suddenly of value to you. Don't expect that to happen anytime soon though.