Do I need to till before putting down new grass?

As ronk mentioned, with new topsoil there’s no need to till it.

If you want to mix in any amendments or anything like that, you could, or if you’re only going very shallow with top soil, it wouldn’t hurt.

I killed and tilled mine when I planted, but I was not bringing in any soil, and needed to really break up the compaction to plant the seed.

Are you going with sod or seed?

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If it is clay then tilling clay doesn’t do anything

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Sure it does. I clogs up the tiller blades........

In Baton Rouge I put down St Augustine sod in 3 different parts of the yard. Two areas were to fix tree removal damage and were not tilled other than to fill ruts. The 3rd area was compacted by decades of foot traffic. It was tilled 6", raked and rolled before laying the sod. All 3 areas look good a year later, but the tilled area looks the best and has the most lush growth.

Note about lawn roller. You can buy a hand roller for about $100. Yes, hand rolling is hard work/exercise. You need it after tilling to settle the yard surface. You may need it after laying sod when edges and corners pop up.