Before you do anything else, get it in writing from your landlord that they are willing for you to have a dog. Any good rescue will want to see this written confirmation as part of the homecheck process and if they don't you should walk away.
You should also think about your garden - it should have secure fencing and many rescues will specify a minimum height (obviously dependent on the age and size of the dog you end up with).
Give some serious thought to the finances of owning a dog. As well as the adoption fee, there will be the costs of buying :- collar, lead, possibly a coat, grooming, professional grooming/clipping (for some breeds), bowls, harness or crate for when out in the car, dog bed for home. You will need to budget for food, regular worming and defleaing treatment, full insurance (this is vital). Even if you have full insurance, some policies will expect you to cover the bill before getting it back, and you won't normally be covered for minor conditions (there's an excess usually) or regular things like vaccinations, worming, dental work etc. Basically, you should be able to access emergency funds e.g. via a credit card for emergency treatment.
For holidays, you will need to cover either the cost of boarding kennels, or a home boarder. If you go out to work, this need not be a barrier to adoption, but you will be expected to arrange cover such as a professional dog walker. Typically, you would be looking at around £10 for each walk for a dog walker with public liability insurance and CRB checked status.
The rescue will need to see that EVERY adult in the house is fully signed up - including your DP or DH. If you are a LP, then they would just expect you to show what you would do to cover if you were ill/unable to walk the dog.
Having a young child is not a barrier to adoption for all rescues, but they will expect you to show that you are willing to keep dog and child closely supervised at all times when together, and that DS is taught very clearly that dog is not a plaything, and is not to be ridden, squeezed, poked, prodded or hassled. Dog should have a secure sleeping area which is respected by everyone in the house and this space is sacrosanct. The rescue may ask if you are planning any more children. They are not doing this to be nosey, but because the birth of a second child, particularly when first child is a toddler, is one of the most common reasons for rehoming a dog.
You should be realistic with the rescue about how much time each day you will spend walking the dog. How will you combine walking the dog with a toddler who may not be so keen, especially when the weather is cold, wet and it is dark outside? At this time of the year, it is lovely but in January/February when it is inches deep in mud, cold, drizzly and you have a streaming cold, not so much.
Think about lifestyle. Are you a manic housekeeper who would be worried about dog hair and mud? Are you already someone who spends lots of time outdoors ? Do you have good, convenient dog walking areas near you, such as parks, beaches, country parks, countryside?
You should also find a vet practice near you, and it is also a good idea to take dog to training classes, such as KC Canine Good Citizen. Great fun and a fab way of bonding and getting the basics right. Usually cost about £40 for a series of around 8 classes.
If you can show that you have given careful thought to these sorts of issues, then you'll be fine. Dogs Trust is an excellent first stop as an all breed rescue or there may welll be other smaller rescues near you. If you neeed any help with choosing a rescue let us know and we can point you at a suitable one.
Good luck! [bugrin]