The combat chapter in the players handbook does a pretty good job of explaining how a turn works. Basically you have your action, and your movement. You can also add small amounts of flourish as you like. There's several types of actions described in the phb, but a spellcaster will usually cast a spell. If a spell says it takes one action to cast, you can cast it immediately using your action.

If you have a spell or feature that uses a bonus action, you can cast it in addition to a regular action. That's how a ranger can hunters mark and shoot their bow during the same turn, or a rogue can hide at the end of theirs. You can only use one bonus action each turn, and if you cast a bonus action spell on your turn, your action cannot be a spell (though you may use a cantrip.)

A spell that's cast as a reaction can be cast whenever the action it's trying to affect happens, even if it isn't your turn. You only get one reaction each round.

Spells that have a casting time longer than one action, like one minute or ten minutes, require several rounds. Each round is considered to be six seconds, so casting a spell that takes one minute will take ten rounds.