Try La Praille or Balexert. Cheers.

Welcome to Geneva. 

France is cheaper than Switzerland, so it's a good idea to drive (once you have a car) over to France for the weekend to stock up. Just be aware that there's an import limit, especially on alcohol and fresh meat. I'm not sure what the exact amount is but you won't get in trouble if you limit it to 2-3 wine bottles and around 1 kg of meat. 

The nearest French supermarket close to Eaux-Vives is the Intermarche' in Douvaine. But without a car that'll be difficult to reach. 

In Eaux-Vives there's a Migros (at the start of Rue des Eaux-Vives, and a bit further there's a Coop (at Place de Jargonnant, inside the shopping center Eaux-Vives 2000. Personally I prefer the Coop as it's bigger and has a wider range of products.

There are also several smaller shops in Eaux-Vives worth exploring, mainly on Rue des Eaux-Vives and Rue de Montchoisy. 

For restaurants: Da Paolo at Rue du Lac 3 has the best pizza in Geneva and is a great place to start. 

I just came back to mention Les Halles de Rive, an indoor market with lots of fresh produce. Between Boulevard Helvetique and Rue Pierre Fatio (entrance on both streets). 

Hi jhon,

theres natural history museum ,10 M walking from Euax

Vives,http://www.tripadvisor.co.uk/Attraction_Review-g188057-d542797-Reviews-Museum_of_Natural_History_Museum_d_Histoire_naturelle-Geneva.html

your kid will love it & it's FREE, you easily spend couple hours there,and you can stop @ the cafe,it's not bad. 

http://www.newlyswissed.com/?p=10426

this might help also
Dalal S, Feb 25, 2013 @ 23:39

Thanks peoples ..  

We've been in the shopping center already, and the indoor deli / market type place.. Tres bien they were, the indoor market is really good, i could spend a fortune in there on food.!!

Natural history museum too, been there a few times :) 

So plan for tomoro i thinks is find Lidl or whatever it's called, Aldi?  Up near the station, rue de lausanne.. Or take the 12 tram up chenne bourg to the shop up there..  

Can we take tram all the way FR? If so, anything worth seeing there?

You can't take the tram to FR to my knowledge. But you can take a bus. The Y bus does a half circle from Ferney Voltaire to the Airport, (Stop right in front of Palexpo entrance) out to Val Thoiry Shopping centre in France. Val Thoiry has nice proper french prices and has Migros, Sports, DIY, Frozen food, toy stores, Clothes, cafes , Paul etc etc. We usually get stuck there Saturday morning. 

BUS map: http://www.tpg.ch/plans/reseau-urbain

Val Thoiry: Website not working this morning used to be: www.valthoiry.com

The F Buss from Cornavin goes to Ferney and stops near the Big Carrefour.

Alternatively think about a mobility car whilst you think about getting your own.

You can pick em up all over town and theyre not that expensive for a hour or two shopping trip.

The best deals on mobility are from the train station. 

http://www.sbb.ch/en/station-services/car-bike/hire-or-borrow/carsharing.html