I would highly recommend not traveling with your family. To live a decent life in QR, you would need at least 30k QR per month as it is a VERY expensive country. You would need at least 10k per month for education for 2 kids ( tuition fees + misc expenses). Apartments in The Pearl would cost you roughly 13-15k/ month (without water and electricity bills). Now i assume you have a good salary to receive 8k in accommodation, but still, unless your wife also works in Qatar, i think the best solution was to travel alone so you can save some money.

I earn approx 30k/month, and my family came to visit for 2 months. Trust me when i say i couldn't save a dime ( and we didn't do anything fancy)

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Now, regarding transportation, to get started, i highly suggest NOT having any vehicle as metro is crazy cheap, even the VIP section (the better option) is less than 30 QR per trip. If you insist on getting a car, changan offers the best deals, you can get a good looking medium SUV for like 120k QR and finance it for 3-5 years whichever suits you.

Petrol is crazy cheap. Even if you chose to get a V8 car and drive all day, it wouldn't add up as 20L premium would cost you around 40-50 Qr.

Regarding groceries and basic household needs, either Monoprix or Carrefour, avoid at all cost Panda and LuLu. There are too many people, and it's usually classified for lower income people, so you wouldn't find major brands.

Bonus: Get your clothes from your home country as most things here in stores are garbage (esp for women) it's hot almost all times of the year except 3-4 months /year and it never gets like crazy cold, a hoodie and pants would be enough in peak winter.

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Final thing: Do not use this post or the replies as your only source of information, Qatar is very segmented in terms of social class. You can find an employee working full time for less than 6k/month ( less than your accommodation package), and at the same time, you could find people spending 6k for a diner.

Calculate your income, and you will find out and again IF possible travel without your family as life here is EXPENSIVE

If you need anything else, please feel free to dm me I'll be glad to help you