Rescues often have puppies but you might have to wait a bit if you want a specific breed. Most breeds have their own rescue that only rehome that breed.
Is there any reason why it has to be a puppy or a bichon? They're cute but not always the easiest dogs in my experience but I do like bigger dogs so that might just be my personal bias.
I'd say do a lot more research about having a dog in general and about the breed you're looking at. There's some guidance earlier in this thread about making sure you're not buying from a puppy farm. If you're not really careful when getting a puppy you could end up with a dog with serious health problems plus you'll propping up a dreadful and cruel trade.
If you do decide to buy a puppy then the kennel club and champ dogs have puppy and breeder lists. Steer clear of gumtree and online pet selling sites and never buy a puppy online and meet them halfway or have the dog delivered to you.
For a good breeder, you'll probably be questioned to make sure you're suitable and you might end up on a waiting list. You need to do some research on the breeder, meet them and ask if you can see the mum and the dad if possible (and ask to see them every time you visit). Ask what health checks they do before breeding, what experience they have, will the puppy be health checked and vaccinated before you get them. What age can you have the puppy as this shouldn't be too early.
Have a look at battersea and dogs trust for rescue dogs as well as breed rescues. I'm not sure what other rescues there are where you live.