It varies by vaccine. Some are injected into muscles ("intramuscular"), some below the top layers skin and into a fatty layer that is rich in blood vessels ("subcutaneous"), some are oral, and one type of flu vaccine delivered via a nasal spray. If it is intramuscular or subcutaneous, the vaccine will quickly get into the bloodstream via tiny blood vessels in the muscle or subcutaneous tissue. An oral substance has to first go through part of the digestive tract, including the very acidic stomach, before it is absorbed into the bloodstream. A vaccine that is spray into the nose can be absorbed into the bloodstream from the nasal mucous membranes.
I'm not a big Wikipedia fan because it is completely open and subject to change, but it has a page on injections