Lawns are one of my favourites. The best thing for them is regular mowing. I used to sell/service mowers and had a test lawn out back. Neighbours used it for foootball and always commented on it's finish. Dont walk on the lawn during bad, wet spells - kills grass. In spring, April, use B&Q lawn weed & feed or similar. Theirs' is quite a good price. Do what it says on the box. Ideally, you want regular rain thereafter, set the mower to about 30-40 mm cut height and mow as often as you can. Not just once every few weeks, I mean regularly. If the lawn has not had it in living memory, scarify it first. This is a labourious task of raking all the thatch and moss out of it. This also opens the ground a bit, allowing new growth.
Scruffy lawns benefit from a second weed & feed a couple of months after the first application. BUT beware, this artificial boost demands a price, I have found that after the drugs, the lawn will want further TLC; mowing, rain, hand weeding etc, or it will suffer in following years. If it is really bad, a new lawn may be the easy answer, ish. Get rid of all the grass top, prepare ground and get expensive lawn to put down. I have never done this, it's only grass! I have recovered what's there. Much cheaper.
Last note, old cylinder mowers will offer the best cut, hovers are easier to handle, rotarys will cut wet grass too. I have a rear roller Honda rotary with grass box, in my humble opinion, the best mower available.
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