A large needle is important, and it needs to have a sharp point, not the type with a blunt point used in needlepoint.
A t-shirt quilt is a challenge to get a needle through -- it is thick and maybe you have used an iron-on stabilizer. Ordinary quilting cotton and batting are much easier. In any case, something that works for my mom's quilting group is pliers. Their utility quilts often have sheets or a thick fabric instead of batting, and they struggle to get the needle through. A pliers gives the grip needed to get the needle through. Mcfay is right about making each stitch a 2 step process. A thimble is another valuable tool that gives you "push power".
A thin crochet thread will be easier to pull through than yarn. If you are concerned about strength of the thread, go up and down twice in the same area before tying.
Another option is to use your machine for "tacking". You still have to be able to get the quilt small enough to fit in the "harp" space, but it is much easier to manage than quilting. Just do a strong stitch in one spot (where you would have tied). Make sure you have pinned at or close to each spot while the quilt is spread out on a flat surface before beginning the sewing.
I have tied many quilts, and I like to tie at the seam intersections (tougher than in the middle of the blocks). It helps me if I can spread out the work, so that my hands can rest. I usually only work about 1/2 hour at a time.
Best wishes!
Dayle