My new lawn looks like a grazing paddock

I'm not winning. I dug over the back garden last year, and had a half hearted (lazy) attempt at levelling it before seeding it.

I did it all quickly and on the cheap (at the time I was skint due to previous boss robbing us).

Now my new lawn is lumpy, and the grass is awful for a lawn. I'd be happy with it if I was grazing livestock, its lovely lush but very coarse grass.

I'm thinking I'm going to have to start again from scratch, killing off the new grass, skimming it, digging it over again, and this time levelling it properly, before either seeding it with good seed or putting turf on it.

Lessons I've learned from this:

1. Don't do the job in a rush. Yes you might want quick results, but you want something you can enjoy for the long term and a bit of extra time and effort up front is a small price to pay for that.

2. Check what you're buying. If the lawn seed packet says 'hardwearing' or 'utility', or words to that effect, translate to 'possibly ideal for grazing cattle/horses/sheep but awful for lawns'.

Lessons I haven't learned (and am hoping for some advice on):

1. How can I level and firm the ground (without compacting it) ready for reseeding (or laying turf)? Heavy machinery is not an option, as there is no access to the back garden except through the house.

2. If I choose to lay turf, how will I know I'm not buying livestock grazing quality as opposed to a lovely mat of fine grass?

3. When should I be thinking about doing this?