Photoshop doesn't handle vectors as well as something like illustrator but it can do them. You can use the pen tool set to shape and draw out the vectors that way. It's been a while since I used PS for vectors but you can add a line stroke, gradients etc. to any shapes you create. As long as you leave them as paths/shapes they should scale as a vector. All text in photoshop is vector based so as long as you don't rasterise it, text should work too.
Photoshop vectors aren't perfect, there's elements of some of it that aren't truely scaleable but if your logo isn't that complicated with patterns, drop shadows etc. then a photoshop vector should be good enough.
Edit: I forgot to add, if you're ok with using the pen tool in photoshop then illustrator isn't too bad for simple graphics. I've only had to use Illustrator a couple times and it's taken longer than it would do in photoshop but I've managed to muddle through each time ok.