Technically, small dogs (like Yorks) are allowed in SOME cafes, most supermarkets and grocery stores. But in reality people do not appreciate it, especially old people and parents with kids. This is the mentality and you should be prepared to get looks and negative response in some cases. If it's a puppy, who can sit in your coat or in your hands while you shop, then there would not be much of the problem. Streets, beaches, parks and forests are absolutely open.
Bars, clubs, cinemas and restraurants do not allow dogs in. If you use public transport, you would need a muzzle, unless it's a small dog that you can hold in your hands. Keep in mind that every dog MUST be kept on a leash in public places, but you can allow it to run free in the small forest (which is almost in every are of Riga).
Try to keep the dog out of places where children are, because you will definitely get angry looks. Such places are all playgrounds, areas close to schools and kindergardens, etc.
There is a very good chain of pet shops called DinoZoo. They sell pet stuff, toys and professional pet food such as Hills, Royal Canin, ProPlan, Ontario. The full list is available hereÂ
You can get their member card and get discount and bonuses. Also they have sales and offers quite often (2 for 1, for example). The other good thing about this card is it gives you discount at most vet clinics (you can ge the list of them in the store).
Overall people are pet tolerant, there are many dogs around and most of them feel quite comfortable and free. Some will ask you if they can pet it and most will not freak our if it comes closer 
There are also yearly dog and cat exhibitions, where you can come and meet other pet lovers.