Avoiding milk and egg is really hard to begin with, but you soon learn which meals work, and figure out which types of biscuits etc are OK too, so it does get a lot easier until you don't really have to think about it much. I cooked dairy and egg free for the whole family, which is much less effort than making two meals.
Good options are: cottage pie, spag bol, fish pie made with soya milk instead of cow's milk, chilli con carne, roast dinners, chicken fajitas, etc. Virtually all dried pasta is dairy and egg free, and most sausages and all fish fingers I've found.
For puddings, rice pudding (make your own with soya milk), any kind of fruit crumble made with dairy-free spread such as Pure or Vitalite, Birds custard made with soya milk (or buy ready made Alpro soya custard). I really didn't enjoy the taste of soya yogurt, but Alpro do chocolate and vanilla desserts which are really nice.
It feels imopssible at the beginning but it does get easier! Good luck x