“I'll go pick up all the stuff we need.”

You are working on a school project with a group of other students. You need to buy some supplies for the project. You want to offer to go out to get them, so you say:

I'll go pick up all the stuff we need.

I'll (do something)

You use "I'll ___" when you have just decided to do something. You use "I'll ___" to announce a decision. For another example:

Sure, I'll add him to the distribution list.

go (do something)

In casual speech, you say "go (do something)" to mean "go (somewhere) and (do something)". For example:

Do you want to go see a movie?

This means to go to a movie theater and watch a movie.

Could you go find me a screwdriver somewhere?

This means to go to the place where tools are kept and find a screwdriver for the speaker to use.

pick up (something)

The phrase "pick up ___" can mean to buy something at a store. When you use this phrase, it sounds like you are buying something quickly while you are on your way to somewhere.

stuff

"Stuff" can mean almost any physical items or material. It's similar to the word "things", but "things" are countable and "stuff" is not. So "stuff" is better to use when you're talking about a group of things that are different from each other - different sizes, shapes, etc.

"Stuff" is especially used to refer to things that you buy or own:

There's a good chance that you'll end up buying a lot of stuff while you're there.

Did you get any stuff for dinner tomorrow night?

I have way too much stuff in my apartment.