It's not THAT bad - I have rampant brambles in my neglected garden, and it's a lot bigger!
OK, nuimber 1, you might need to build a retaining wall and add topsoil to get a very flat lawn, I wouldn't recommend that, as you will get water running off towards your house, which may well lead to a damp problem. Maybe a series of terraces so you can have flat strips of lawn?
2, If the grass really is that bad, then maybe you'll need to dig it up and start again, either with seed or new turf. You need to decide which PDQ, as I wouldn't recommend trying to start a new lawn after the end of April, as the weather tends to warm up and not be as wet, you'd spend a lot of time watering it to make sure it didn't dry out. Do you have a lawnmower? If so, make sure the blades are on the highest setting, and give it a once-over. If it looks terrible, some bits down to bare mud, and other bits are long, I'd say dig it up, level the ground and put new lawn in.
3 Ivy is a bugger to get rid of. You need to bruise the leaves (try rubbing gently between your hands, without pulling the leaf off the plant), this gets rid of the waxy coating that's on an ivy leaf, which means that any weedkiller you spray on it will actually be absorbed by the plant. Make sure it's not a plant that belongs to next door, and they are happy for you to get rid of it, next door has ivy that comes over the top of the wall, I just chop it off whenever it comes over, the chap next door would have a hissy fit if I killed his Ivy! I would use a glyphosate weedkiller to get rid of the weeds, including thistle, dandelion etc, and once they're dead, use a long weeding knife or bulb trowel to get rid of the roots. If they're in your lawn, fill the hole with a mixture of sharp sand and grass seeds, to fill the hole, improve drainage in your lawn (unless you've got sandy soil anyway, then I'd use compost & seed), and the seed should germinate in a few weeks so you can't see the hole where the weed used to be.
Getting rid of weeds can be a slow process, I use a mixture of glyphosate, digging and bloody-minded stubbornness.