Quick answer
Добро́ пожа́ловать!
dobró pozhálovat — the sign at every door and border
The options
Добро́ пожа́ловать!Welcome! — signs, speeches, formal
Проходи́те!Come in! — what a host actually says
Раздева́йтесь.Take your coat off.
Чу́вствуйте себя́ как до́ма.Make yourself at home.
Сади́тесь, пожа́луйста.Please sit down.
Shoes off
At the door you will be handed та́почки — slippers. Keeping your shoes on in a Russian flat is close to unthinkable, and the offer of slippers is not optional politeness.
Not сиди́те
“Sit down” is сади́тесь. Сиди́те means “stay sitting”, and in the wrong moment it sounds like an order.
What you hear back
Спаси́бо!Thank you!
О́чень уютно у вас.Your place is lovely.
Э́то вам.This is for you. — handing over a gift
Typical mistakes
Добро́ пожа́ловать to one guest at your door
Проходи́те!
The formal phrase belongs on a sign, not in a hallway.
Keeping shoes on indoors
Take them off
Not grammar, but it will be noticed faster than any mistake on this site.
Check yourself
A Russian host opens the door. What do they most likely say?
Проходи́те — “come through”. The formal phrase lives on signs.