New lawn, clover and unknown weeds. (newbie)

Hi, I am finally going to try and sort my lawn. It had a lot of moss which I've used a moss and weed killer on. It's went black and raked it up. The lawn also had a lot of clover on it. I've cut it very low to see what it's like now and there's still alot of what I would describe ve as vine like weeds.
Im not after a bowling green I just want some thing that's actually grassy and looks half decent. I've attacked the pics of what I mean. 

What's the best plan of attack from here? A treatment or kill the lot and start over? I'm new to the gardening world but have spray/pump if liquid is needed to be applied.

Thanks