Complete lawn re-do. Any way to firm up the ground before re-sodding?

A soft lawn would be a rarity. Where are you located?

Always remove the grass. Whether or not you rent a sod cutter or round up and rake you want to lay the new sod on bare dirt.

If it is dry durning the summer I wouldn’t mess with it too much. Especially if you plan on bringing in new dirt. I have a spot that stays wet for up a week after a decent rain. It only happens during the winter.

I’d just round up the lawn and wouldn’t worry about raking up the dead stuff. Wouldn’t want some zombie grass hanging on and fighting through.

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last time I added some fill it was full of weed seeds or whatever

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What kind of "fill" was it?

I think you would need to use some good top soil and/or mason sand (potentially both mixed together) but I'll defer to the experts (ronk, Craw, etc.). Avoid river silt / sand at all costs.

Use masonry sand to level out and you can sprinkle a thin layer of compost, peat moss, or garden soil under the sod.

As others have said stay away from getting a giant truck full of river silt.

I’ve always removed the old grass.
Not sure what kind of soil you got, but stick with good top soil from a reputable landscaper or materials yard.
If you want to compact your topsoil, you can rent a jumping jack, compactor from Home Depot tool rental.Its a good idea if you want to put sand on top of your topsoil for drainage.
Hope this helps.

No personal experience but a neighbor did this several years ago.

Guys came out with weedeaters and they must have had some serious lines on them since they cut the lawn down to the dirt and removed the grass. Then they tilled the topsoil about 3 inches deep to loosen it before leveling the soil. Then a couple days later they laid new St. Augustine sod and had him water it daily for a few weeks until it set down good roots.

Looks great