Quick answer
Извини́те
izviníte — to get attention and to apologise — the same word
The options
Извини́те.Excuse me / sorry. — polite, both uses
Извини́.Sorry. — to someone you call ты
Прости́те.Forgive me. — a shade more serious
Извини́те, пожа́луйста.Excuse me, please. — softer
Скажи́те, пожа́луйста…Tell me, please… — the standard opener for a question
Getting attention versus apologising
Извини́те works for both. For a real apology, прости́те carries more weight — use it when you actually did something. For stepping on a foot, извини́те is plenty.
The polite question opener
Almost every question to a stranger starts the same way: Извини́те, скажи́те, пожа́луйста, где метро́? It sounds long, but it is the standard and it works every time.
What you hear back
Ничего́.It’s nothing. / No problem.
Всё в поря́дке.Everything is fine.
Да, пожа́луйста?Yes, how can I help?
Typical mistakes
Извини́ to a stranger
Извини́те
The -те makes it polite; without it you are using ты.
Я извиня́юсь.
Извини́те.
Я извиня́юсь literally means “I excuse myself” and is widely considered poor style.
Check yourself
You want to ask a stranger for directions. How do you start?
The polite opener is Извини́те, скажи́те, пожа́луйста — the whole phrase is standard.