1) Some things you absolutely need like Fleet logistics, planet settling techs and the upgrade to certain types of ships based on your battle preferences. If there are multiple stars you need to jump to them as well. Basic repair and air defense can figure in here as well.

2) I would consider those techs that allow you to get resources quicker, build things for less, cause population to grow are kinda in the second group.

3) Then there are the rest. It is nice to know when someone is jumping at you. There are the warfare techs like shielding and repair that are great to have as well.

4) Some of it is just candy for the last grouping like a Novalith Cannon.

For my two cents as a relatively new SP person I try to do most upgrades in the above order after I have sufficent fleet and types of ships for my purposes. Extra cash/resources available? Upgrade.

My last game was dramatically different than the previous as I just basically built fleets to Assailants with a few Caps/Destructors and took/held planets. Cash flow was not as great but I had fun.   

Multiplayer I'm sure is entirely different.

/agree PaulTa

Everything so far has been good advice, I agree with everything.

Here is just something I do:
Ships, colonization, long range and trade ports are all techs I get when I can/need too. Shield and weapon upgrades I do a bit differently. Since teching each fleet level has game long effects on your economy I am always reluctant to increase those too early. So instead of teching up a fleet level I will start teching my ships before I HAVE to upgrade my fleet. Usually I will start the shield/hull/armor right before going to fleet upgrade 2; I will do weapons right before fleet upgrade 4 or 5 and then usually one tier more each fleet level I go up.

With weapon upgrades I usually pick and choose which to upgrade based on on what ships I am going to be using/are using more. So as Advent, plasma is last to upgrade because I dont get HC's till later and there really is no other ships that use it predominately
and beams are first because I have tons of Illums and bombers.

Not sure if anyone else does the same thing, but it seems pretty logical.

As Sixhits says: One thing I learnt from other RTSs is that you shouldn't buy research upgrades for units that you don't have any of. Sounds obvious, but it's worth stating as money can be wasted on things that you think "oh, I'll be getting some of those in a minute", but it turns out it's a long before you do and that cash is useless until then. However, this is less of an issue in Sins as many of the researches affect lots of different units.

With the resources boost though, I would obviously concentrate on what resource is your bottle neck, rather than purchasing them indiscriminately as they can cost a lot. If all three resources are about even (not in terms of size, but in terms of which one you are always waiting for more of) then you need choose between going and taking more planets or spending some cash on researches.

Not sure how wise this is so take it with a pinch of salt, but I usually snap up a few of those cheap, early researches if I'm running short on metal. It can get frustrating and inefficient to have to buy expensive blackmarket metal, so sometimes I figure I'll just get them out of the way while certain resources in my economy are floating. This is especially true for (I'm playing Advent at the moment) the defense researches that you need to allow the Iconus Guardian, and also the 3 levels that speed up research, as they are quite cheap too and can help. Then again, I wonder whether buying ships that are cheap in metal would be better.

Lastly, they are useful if you are generally floating and your shipyards are busy (rare I know). Overall you need maximise the bonuses of research combined with fleet/economy choices. Trim the fat from both and make the most efficient use of resources.