K.S.
I'm not sure if you've tried this, but we found canned veggies the easiest at first. I did look for the low salt ones, since they are pretty high in salt. Frozen and fresh were just really tough, even when cooked well. It was a very long and gradual transition.
Meat was even tougher. The key to it all was to keep trying. Over time it became more and more familiar to my son. We used babyfood meat for a long time. The next step was a sort of mechanical soft diet. Well cooked meat that was ground, but not pureed, with lots of moisture in it. Fine ground beef was the next step. Then very soft meats such as Rotissure chicken in very small pieces. At 3 1/2 meat is still not a huge thing for my son, but we've made lots of progress from baby food.
If things do not progress, a therapy evaluation can help with any sensory issues. Good luck! You are just entering the true challenge of getting your child to eat!