If the lawn is compacted then it may help to dig over the area slightly, and also this is especially useful if you intend to grow root vegetables.

As Nicky says it would be a good idea to break up the soil below the grass with a fork so that it drains properly & so that roots can reach down. Worms will also be able to come up which will help aerate the beds & make sure drainage is good.

As for compost you could use general all purpose compost to which you could add some garden compost, if you have any. You could also chop up the grass you intend to remove & put it in the bottom of your beds before adding the compost.

Good luck with your new raised beds! :-))

If you dig it over when its workable? The frosts will break it up for you and perhaps add some grit if its solid such as clay?