Do stash size increments carry to new characters?

Just started playing, the 10k to increase my stash size appears very costly right now.

Does that increase then work for all characters on my account or will I need to spend the 10k+ for each character that wishes a larger stash?

The Stash is shared between all of your (non-hardcore) characters. Capacity upgrades are shared as well.

That said, early on stash capacity upgrades are an extremely inefficient use of your gold. You're better off making an additional character to act as a mule to hold on to that cool axe you want to give to your barbarian someday - it's slightly inconvenient to move stuff around, but compared to the inconvenience of raising 10,000 gold at this point in the game, it's definitely the better choice.

Every account has two stashes that are shared between characters (one Hardcore, one Softcore), so any stash expansion upgrades affect all of your characters of the same type.

Again: Hardcore and softcore have separate stashes, so upgrading a softcore stash will not increase the items available in hardcore.