I personally would get rid of all the weeds with round up and get rid of what you don't want hence the stones etc. and I would sod the lawn it looks small from the pic and won't cost that much I did my entire lawn and it was a few hundred but so worth it! :) you can make that look awesome with some beds and mulch and perennial bushes that flower and even a pond, I put one in the front its so therapeutic ..Good Luck

If your yard is really 600 sq ft, your best bet is getting sod. You could have a nice lawn by next week for a couple hundred bucks that way. Just clear out the weeds, rake it out and roll out the sod. If you want to go the seed route, there are about 1,000 treads on here detailing how to do it. The best time to seed in your area is after the summer heat breaks, so you have plenty of time to read up.

I am not familiar with your area and what types of grass is grown and readily available, so I will let someone else comment on that. I seeded Bermuda, im in texas, about 2 years ago when my house was built. I started off with rye grass during the winter, but since it did not take everywhere, it resulted in areas where there were washouts. I did not level my lawn again before planting Bermuda, wish I would have, but it is not all that bad for me. I am currently taking my time in leveling the lawn now with a very good amount of grass. I am doing it in small sections as I have over 3/4 of an acre to maintain. So back to your situation and what I would do. If you think you have bad soil, maybe consider doing a soil test. While the soil test is being done, spray the area with round-up if you are trying to get rid of whatever you currently have. If you do not want to do that, then bring in topsoil, and then get someone to use a box blade to smooth the area and then seed whatever and whenever is recommended. I will strongly recommend not tilling. I have tilled in certain areas, where I though the ground was too hard to seed and it did result in a bumpy yard. Tilling also roots up new weeds that you currently do not have. If you do some searching here and other sites, you will see some good posts on leveling, prepping and seeding your yard. Hope that helps as a start for you. To keep this post short, I will try to answer anything else if you have specific questions.

Yes, we can see you are inexperienced, but that's okay. It might be worse to come in as I did thinking I knew it all and being dead wrong about most of it. We need to know where you live to know what kind of seed you need. Some seed must be done in the hottest time of year and the rest should (must) be done in the fall. You rototilled it, right? That was a mistake. If you can walk on it to make it sort of level, do that. It needs to be very firm to do anything. When you get it sort of level, then find something to drag behind you to help level it. I've used a piece of chain link fence with some boards on it for weight. Then water it deeply once. Watch for low spots where the water collects. You want to fill those later after the soil dries enough. Let that dry out completely (need to know where you live again), water it deeply again and let it dry. The idea is to get the soil to start building a structure that you can walk on without sinking in at all. It should so firm that you cannot see foot prints at all when you walk on it. Part of the problem is the buried organic matter. That stuff will completely disappear as the microbes in the soil process it. As it disappears it robs nitrogen from around the soil and leaves voids that will result in lowering the surface. This is going to be a bumpy surface for several years, but it can be leveled later on. The other thing is the manure will stink for the next few weeks. Never use manure in your garden. Ever. If you buy manure, pile it up, cover it with dry leaves to block the smell, and let it compost. Eventually it will become very nice smelling compost.