The first item I would try to research for is a SCOPE to equip to the sniper (with squad sight). Then get an armor set, whatever you can afford but carapace if you can. Then beam weapons, plasma if you can, but whatever you can afford.
Really what you need to concentrate on is satellites. The first thing you should build is a workshop to bump up your engineer count, then a power generator, then another satellite uplink. Simultaneously build satellites (3 at least, but 4 if you can afford it) while these buildings are being completed. These may require you to excavate so that your power generator is near a steam source.
All this stuff cost credits and time and are more important the research items at the beginning. To afford this I pretty much sold everything on the gray market which is why I did almost nothing in the first month. Although I saved enough stuff to research I mainly tried to keep my soldiers alive and worked to mitigate panic instead of buying items to equip. Check out the below video that I found help, it shows a tip to launching 4 satellites in the first month, not easy but doable.
http://www.youtube.com/watch?v=dpVrQDw3pds
One other tip I found very helpful, don't launch your satellites until a week before the council report. This way you can optimize the encounters, reducing panic your squad instead of valuable satellites. It takes time/credits for them to go up so save them until you need them. This can make the missions more difficult since you will be playing difficult/very difficult missions exclusively.
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