Thanks for crediting the GOP with "upping" the minimum wage but federal wage minimums are set under the rules of the Fair Labor Standards Act.
Most states have their own minimum wage and the higher rate applies to employees of that state. Some states set their minimums using a cost of living formula or inflation so rates fluctuate annually. Some states pay less than the minimum wage for the first 90 days of employment, considered training periods. Some states also allow employees under 18 to be paid less.