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Hi Natalia, welcome to Dubai.
There are a number of different furniture stores in the malls depending on price - not too expensive are Homes R Us and Home Centre then there are more upmarket ones like The One. There is also Dragonmart for cheaper things - they sell everything you could possibly think of furniture/ bedding/ lights, curtains, clothes and lots more that you will never need in your life!
There are some great delivery stores www.choithrams.com - not the cheapest but generally good quality and geant.com is a large supermarket store. www.qualityfood.ae are very good for fresh veg and meat.
Most of the large supermarkets like Carrefour and Geant will supply the necessities of life.
It really does depend where you live in Dubai as to what is available closest to you - but don't worry there is a reason Dubai is world renowned for its shopping! Another good online one is souk.com for electricals.
I am sure there are more but can't think of anywhere else at the moment.
Good luck with everything I'm sure more people can help with their favourite websites!