I have moved several times and for the most part, the companies are all the same. The company can have a great reputation or national name but it all depends on which guys they send out to service you, I have had great guys who are fast and reliable and then some lazy ones who have damaged my furniture and who showed up with trucks not big enough. Tell the company you want to be assured they are sending you their best men.
To save money, do as much as you can by yourself or with some friends. You could make them dinner as a payback, offer to babysit or something. Nobody likes to move but if I had a friend who was a single Mom, I would definitely help her.
If you are able to move in some things before the day you want the movers to arrive, that would cut down costs. Make as many trips as possible moving things you can carry and just leave the heavy stuff to the movers. They can have all your furniture packed up in an hour and ready to go...and you usually pay by the hour. Make sure everything is ready to go so there are no packing or preparation delays. Ask the company what you need to do regarding getting things ready to go like your clothes in the closet (they can pack with hangers on) or securing the drawers on your dresser, etc.
I don't know how often you have moved so I hope this doesn't sound elementary, just offering tips.