I played them in order, skipping the first 5. Before Touhou, I'd never really played any shmups, but after a lot of practice I got the hang of it.
So grab Embodiment of the Scarlet Devil (Touhou 6)! There's no shame in playing on Easy with continues and 5 lives at first, but try to get out of the habit as soon as you can. Use the practice mode a lot to get the hang of each level's "flow," and pick up a game controller if you don't have one already. I have the Logitech DualAction, which is only $20, but any (with sticks) should do you well.