I usually add new yarn by knitting two stitches with both old and new yarn held together. Also, I try to do that in a place where there’s less lace pattern like on a garter stitch edging, on a filler stitch sections, etc. Once the shawl is blocked you can hardly see them.
Some people will just trim the ends then, but I additionally wave the ends in as well. I separate the yarn end into two thinner strands and thread one strand on a small sharp needle. I then weave it in one direction following the stitch path of the pattern (for an inch or two). Make sure to take the needle through the middle of the last knitted stitch before you stop weaving – this splits the yarn and locks it in place. Then do the same with the other yarn tail, except go in the opposite direction.
You don’t have to split your yarn ends, but doing this prevents the lace looking too thick.