Make sure the type of dog you are applying for actually suits your home and lifestyle.
The tv programs NEVER show the full details on homechecks and the rehoming process, ignore that.
GO to your local rescues, be a real person NOT a set of details on a form.
Make sure your property is suitable - fencing is a BIG deal so make sure thats good before you apply.
Speak to breed rescues, talk to breeders too about particular breeds and express an interest in an older dog - many breeders 'run on' puppies to around a year old to see if they will make the grade for showing/breeding, then pick.. rather than picking at tiny puppy stage - but these dogs almost certainly will NOT be offered to complete strangers so you need to get to know folk.
Go to shows like Crufts, and events like Discover dogs and meet breeds in person and the people that own them - they'll be involved in the breed clubs (the people who bring their dogs to discover dogs will certainly be breed club members or close friends with someone who is) and these are the people who know about older dogs needing rehoming, run on pups etc.
DO NOT get an overseas rescue because they will flog you a dog from photos you just cough up the transport costs and pick up from a motorway services carpark - these outfits are AWFUL and no matter how many people pop on and say their dog is wonderful I can show you multiple cases, over many many years where there is no back up, no safety net, dogs are not suited to be in pet homes, dogs are aggressive, bitey, terrified, escape artists and absolutely require more work than a puppy would.
Also avoid the free-ads - good breeders do not need to advertise like that. Those advertising are unlikely to be good breeders.
Another route IF you like greyhounds is to find trainers (they're based at tracks) and ask them if anything is coming up for retirement - you won't get a lot of choice, you won't have any rescue back up - but theres a strong chance you will get a nice adult dog who needs to learn how to be a dog (not the route for everyone admittedly!) - I love them but they're not the dog for everyone, by any means!