Is a bracelet that is stamped 14k on one piece and 585 on the other real gold?

Yes it is solid gold but not completely pure. Gold is normally alloyed with other metals to give strength, and the amount of gold in the mix is measured in carats. Pure gold is 24 carats, that is 100% gold. Your bracelet is 14 ct, that is 58.5% gold, which is good. A lot of relatively cheap gold is 9 ct which is only 37.5%, but that is still enough to prevent tarnishing and it is sold as gold. Expensive jewellery is usually 18 ct, so 14 ct is in the middle.

If you are interested get a magnifying glass and look for the hallmark, which (in the UK) will give the maker, the town where it was assayed (officially tested) and the year it was assayed.