Most classes I took recommended 300dpi, a lot of software out there has 150dpi as an option to output but I find that too low. If I'm working on something that I think is important, I'll work at 600 or even 1200dpi but this tends to be overkill. It's more important for highly detailed things than for a simple logo with a few shapes and outlined text.
Through the software. I would recommend that you go online and look through some Photoshop or GIMP tutorials, because you really need to learn these things if you are going to be making t-shirts.
If you are talking about resizing an image off of the internet to put on a t-shirt, if your image is only a few hundred pixels in height and width (say 480 x 320), then the answer is pretty much "You can't" ...unless you're planning on printing it out at 1.6 x 1.06 inches.
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