You can always start with your family doctor, from there you might be referred to a rheumatologist, for example, if they think it's Raynaud's Disease or a vascular specialist if they think it's peripheral vascular disease. (although vascular specialists also treat Raynaud's)
If you have numbness and tingling ("pins and needles") you might see a neurologist.
Acrocyanosis is often a benign condition that causes discoloration similar to Raynaud's but Acro is considered largely harmless and there is no treatment other than using hand warmers.
It really depends on your condition. Do you have high blood pressure or high cholesterol? Diabetes? Do you smoke? Any history of heart problems?
(you don't have to answer all those personal questions if you don't want to
, those are just examples)