A lot of people here use either illustrator or photoshop, but Inkscape or GIMP can be a good place to start practicing before going ahead and spending money on that sort of software.

Learning to draw digitally can be a bit of a challenge. Many illustrators use drawing tablets instead of a standard mouse, you can pick an entry level one up for less than a hundred bucks, and they're quite a bit of fun. It's also common to scan a pencil drawing and colour or trace it digitally to make a tee design.

But, you can also create illustrations from scratch using things like the pen tool in inkscape/illustrator - it takes a bit of practice but the results can be really good.

I'd recommend picking a method that feels most natural to you and just practicing until you get the hang of it. It's like working with a new drawing material - there's a learning curve involved, but it's worth sticking with it til you've mastered it.

Coming up with ideas that are printworthy is a whole other skill, but if you can master both of them you'll be set!