First Time Leveling Burmuda Lawn

How big is your yard? If less than 5000sq ft, the using those tools by hand would probably be manageable. I have the top one and it works well - wider the better. Landscape rake is good for the initial knockdown and spread but not as good for leveling. Any bigger than that you'll probably want to look into a pull behind leveler than you make yourself for a riding lawn mower or ATV.

With that said I would say it depends on your problems and your goals. If you have serious drainage problems, big low spots , etc. Then as mentioned already going nuclear and regrading the whole area will get you the more satisfactory results than you can fine tune with future leveling etc. Also allows you to bring in "good" dirt as a base.

If you're yard isn't in terrible shape then. Yeah you can attack it over several years.

The problem with mixed soil and compost is that while it brings in nutrients/organic matter, is that it will as expected decompose over time, so you'll eventually be leveling again. Top soil in general can be difficult to "fine tune" as it's bulkier and more likely to have "trash"(partially decomposed wood etc) but can be used for the lowest divots and worst spots. masonry sand obviously allows you to fine tune, more uniform in composition as long you make sure your supplier screens it appropriately but you use way more than you think.

Have you tried a lawn roller? Maybe that's all it needs

I am paying a company to do my yard. He will wait until June to do it and he will use masonry sand. I have about 5,000 sf and he figured I would need 20 yards of sand. When the sod was planted last year the yard was very wet, so it has a lot shallow ruts in it.