Dyeing is pretty straightforward. Soak it in dye mixed with water for 24 hours and wash it gently afterwards.

You're going to need a bucket or other large container, because you'll probably need your sink in the meantime.

Purchased dye works great but it costs money and might have chemicals. A usual batch of dye can color a whole lot of clothes, so you're in luck if you already have more than just one shirt to do. Once open, it's difficult to store for later.

If you want a DIY non-toxic dye, you can simply use coffee for a light tan shade, or blended beetroot for red color. Lots of other options, but they will be more light than purchased dye. Natural dyes also won't last as long in washing, but it's a fun experiment to make your own. Same procedure. Mix it in boiling water and check that the fabric can withstand the heat (no knitwear). Otherwise let it sit for longer in colder water.

Check the polyester percentage as well. You can't dye plastic.

To make patterns you can simply hang the clothes over the dye mixture and only drop dye the parts you want. The edges will likely not be very straight, but if you draw the lines up with a black marker afterwards it can look really cool still. You can also make the hippie tie-dye using rubber bands to separate the areas. I'm not really a fan of that look, but I'm not going to judge anyone for looking like a stoner. I much prefer to color the whole thing, because it's easier and looks better in my opinion.

If you just wanted it to be white again, you can do the same with bleach, but keep in mind that it's toxic, smells like a swimming pool and bad for the environment. Bleach might be better to do in the washing machine to keep the smell contained. Don't put other dyes in the washer - that won't work well.