If you want a lawn (a large, flat area of grass), there are two ways of doing it. You can sow grass seeds, or you can buy rolls of grass that is growing in a thin layer of soil (called turf) and then lay it in the same way as you would lay a carpet. This is what turf looks like:
Our front yard would be a lot nicer if we lay down in grass
This sentence is not correct. It could be made understandable by removing the in, but still not natural or grammatically correct. More natural sentences would be:
Our front yard would be a lot nicer if we made a lawn.
Our front yard would be a lot nicer if we sowed some grass seeds.
Our front yard would be a lot nicer if we laid some turf.
You could also use the words lay, grow or plant in relation to a lawn. The term lay down is not used about turf or lawns. This NGram shows that make a lawn is considerably more common than lay a lawn, and lay down a lawn hardly appears at all.
Note that, when we use would in a conditional sentence (one with an if), we normally backshift the tense of the verb in the if clause, so make->made, sow->sowed, lay->laid.
This is what the first sentence would be like if we made it unconditional:
Our front yard will be a lot nicer when we make a lawn.