You can take a look at the Rescue Review website. You can ask them if they are members of the Dogs and Cats Home Association. You can ask them if they are a registered charity (not being a charity isn't necessarily bad, but being a charity means they should have a minimum level of governance and a reasonable degree of financial transparency). If they are a charity you can look up their trustees, financial records etc on the Charity Commission website.
If part of their operation is a boarding kennel, check with the local council and ask to see the last three inspection reports - you have the right to under FOI - this is often very revealing.
Can you tell us their name? There are many people here who are involved in rescue and may be able to help.
A good rescue, irrespective of size WILL - homecheck before letting you adopt. If you are renting they will want to see proof that your landlord will allow pets. They will be interested in everyone in your home and will not proceed until ALL adults are completely on board and signed up. They will want to check you understand your responsibilities as a dog owner and the particular requirements of the breed you are adopting.
They will make sure the dog is chipped, vaccinated, wormed and de-flead, and had all necessary medical treatment. They will be able to give you a comprehensive health assessment. Ideally the dog will be spayed/neutered, or if too young they will have made arrangements for this.
They will be able to provide a comprehensive behavioural assessment. Ideally, this should be by the dog living in a home environment with foster carers - this is the gold standard, and is particularly important if you have DC, or are interested in a breed with special challenges. They should be able to tell you if the dog has any issues e.g. separation anxiety, nervousness etc. They should be able to provide high quality behavioural advice FREE OF CHARGE. Walk away from any so called rescue (yes, Many Tears I am looking at you) that tries to charge for this.
They will make sure they will take back the dog if there are any problems in the foreseeable future.
They offer ongoing support and social events for adopters - apart from the social/fund raising aspect, it's great for adopters to enjoy the mutual support and information from these sorts of things, like regular walks, fun dog shows, BBQs, summer picnics, Christmas parties etc. For example, many greyhound/sighthound charities offer safe spaces for off lead play, or specialist training classes.
If they mention Cesar Millan or dominance based training, run for the hills and do not go back.
If they encourage you to go to Dog Training Classes, or make this a condition of adoption, be pleased - this is an excellent sign. A well trained, responsive dog is much nicer to own and to enjoy, boht for you and for eryone around you.