I moved my shop quite a distance with movers and then once by myself locally. Honestly, I would go with movers again. I would move things out of the basement out of concern for my tools, my stairwell walls, and the movers' health. They are going to brute force things out, and something or someone could easily be damaged. But after that, I would trust them to get them into the truck and back out on the other hand.
I doubt they will move things into your new basement. Usually most moving companies state they are only responsible for moving items to a reasonable location. Pack your boxes too heavy and you will find them all nicely stacked in your living room. I think with tools, if there is a garage at your new place, they will them there and call it good. Also, I would highly recommend you have them pack your fragile items that you cannot move yourself. If you pack something and it breaks in transit, you have little to no recourse. If they pack it and break it, you have some recourse.
Cheers,
Chris
If you only took one trip to the hardware store, you didn't do it right.