The key to sewing seams on a t-shirt (or on anything made in a stretchy knit fabric for that matter) is that the stitch needs to be able to stretch just like the fabric stretches.
Zigzag stitches are often recommended for use if you don't have a serger, but what you really need is a stretch stitch. And, many modern machines, even less expensive models have a stretch stitch. If you have icons to show the stitch types on your machine, this is the icon to look for:
If your machine doesn't offer this stitch, you might try looking in your owner's manual for information about sewing with knits...it should tell you the best stitch to use on your particular machine.
If you are used to sewing with woven material (like quilter's cotton), sewing with knits can be a bit challenging because the fabric will stretch while sewing, the edges will roll up, and regular pins and sewing machine needles can cause snags.
Luckily there are tons of tutorials on-line about sewing with knits, so just google "sewing with knits on a sewing machine" and find the tutorial that works best for you. Here's a couple to get you started http://www.jenniferlaurenvintage.com/2014/06/sewing-machine-settings-for-knits-if.html and https://www.seamwork.com/issues/2015/06/knits-without-a-serger (at a glance these both look useful, but I haven't read them, so I can't vouch for their accuracy).
BTW--if you can't find your machine's owners manual, I highly recommend searching for a copy on-line. Don't pay for one unless you absolutely can't find it anywhere for free. Try the manufacturer's website first, and if your particular model is not available, reach out to them and ask for one. Also, lots of people on the interwebs have been kind enough to scan and upload their manuals and will share them for free. I like doing this because I can save it somewhere (on my hard drive, in cloud storage, etc.) and I can always track it down again when I inevitably lose the hard copy version.