The presence or absence of teeth shouldn't affect whether or not your child can advance her repertoire of solid foods. The front teeth are primarily for biting or tearing a small bit of food off a larger piece. You then use your back teeth (as well as your tongue and palate) to do the actual mashing and grinding of the food to get it small enough to swallow. Infants use their gums to help mash their food so that it can be swallowed. For most foods (bread, pasta, cereal, soft fruits, and tender, cooked vegetables and meats) this works perfectly well.
There are definitely some foods that you should avoid giving your child at this stage. Nuts, hard candy, hard raw vegetables, raisins, popcorn, and hot dogs all require hard back teeth to handle them properly, and to minimize the risk of choking. You should not give these to children who don't have their molars, and I recommended you avoid them completely until your child is at least two years old.