First, welcome to the board!
This is a common problem. There are several things you can do. First, I'm guessing that you have a zigzag needle plate. This has a wide opening to accommodate the back and forth movement of the needle. You should see if there is a straight stitch needle plate available for your machine. This has a small circular hole which supports the fabric and lessens the chance of it being eaten.
Next, you need to either hold both thread tails to the back of the machine as you begin stitching or use a leader (aka, startie-stoppie or spider). This is a scrap of straight fabric that you start to sew onto, then chain piece your triangles after. When you get to the end of your chain, snip the leader off and sew onto it so you are ready for the next chain.