Which hardware improvement would give me the best performance?

Unfortunately ddr2 is more expensive than ddr3 typically. There are socket 775 motherboards with ddr3 support, but honestly if you're going to go to the trouble of upgrading the motherboard it would be more beneficial to upgrade to something newer than the 775. 2gb of ram isn't much for most tasks.

If you're trying to play games, the ram is a bit of an issue and so is the video card. If you're not playing much in the way of games or not trying to play any somewhat recent games, then the video card should last you for basic video (youtube) and some occasional light gaming, facebook games etc.

You're kind of between a rock and a hard place (my current specs are similar so I know where you're at).

Here is a decent (and one of the least expensive) lga775 motherboards with ddr3 support. I have this exact board in another system, it's not bad at all. Around $70.
http://pcpartpicker.com/part/gigabyte-motherboard-gag41mts2pt

This is about the cheapest set of 8gb ram you can get (2 4gb sticks) for $58.
http://pcpartpicker.com/part/team-memory-ted38192m1333hc9dc

The trouble is, ram isn't near as cheap as it was a year or two ago. This option (new motherboard and ram) sets you back around $130.

Another option would be this:
PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($64.29 @ Amazon)
Motherboard: MSI Z97 PC MATE ATX LGA1150 Motherboard ($80.98 @ Newegg)
Memory: Team Elite 8GB (2 x 4GB) DDR3-1333 Memory ($57.99 @ Newegg)
Total: $203.26
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-12-11 07:21 EST-0500

For right at $200, you'd get a newer dual core cpu (for the time being, and a great overclocker if you add an aftermarket cpu cooler) - you'd get 8gb of ddr3 ram which will be plenty and quite useful for this processor or an upgraded processor in the future. Instead of sticking with the same motherboard and only moving from ddr2 to ddr3 ram on an old platform, you can use that same money to move forward to the z97 platform.

You needed a motherboard and ram anyway, your current video card will still work with this until you can upgrade it. The g3258 will be just as good if not faster than your current cpu (since you can't move your e8400 to the new z97 motherboard. You'll now be all set up to drop in an i3, i5 or i7 and have a motherboard that will support broadwell when it's released too. For now you can reuse your current power supply (it may need to be upgraded before you get a new video card depending what power supply wattage you have).

I just think for $70 this would be a much smarter move, bring some of the core components current and put you in the right direction rather than investing that $130 into basically just a ram upgrade that's not going to get you very far. This option gives you half of the components you need for a much newer viable system since it gets the ram and motherboard out of the way. Otherwise when it comes time to upgrade again, you're going to be faced with needing a whole new motherboard again, new cpu, new video card etc and the only real hardware you'll be able to transfer would be the ddr3 ram.

There shouldn't be anything else you need for the time being no. I don't know what power supply you have, but most have the proper connectors on them. If you could list the make and model number it would be easy enough to check.

The prices I listed were in us dollars - I didn't know you were paying in euros so I don't know how those parts would translate to your budget both at european prices and in currency exchange. But yes, you got it - option 2 would do all the same things and have you set up for smoother future upgrades. For instance in 6 or 8mo if you decided to upgrade your cpu you could easily drop an i5 quad core processor right into the updated motherboard without changing (or needing more) ram or anything else. Something you couldn't do with a replacement lga775 motherboard and keeping your e8400.

Given your problems, slow program loading etc it's definitely your ram. What happens is a pc tries to load as much currently used info into ram as it can. If there's not enough ram, then it has to keep shuffling info to and from the hard drive while you're using your programs. Hitting the hard drive so often isn't the best for the hard drive and makes things seem extremely slow in usage.

For instance I'm not using ddr3 at the moment just ddr2 and I'm using the same exact cpu that you are. I have 21 tabs open in chrome, solitaire open, task manager, a hardware monitor program and 50 tabs open in firefox. My ram usage is 5.13gb out of 8gb. Everything opens up as fast as I click on it, no lag. I'm not running a game, but my point is even with a few browsers open it can utilize a lot more ram than you have at the moment.

I wanted to upgrade my cpu this year and to do so means a new motherboard and new ram (ddr3). If I was in your shoes with limited ram, I'd have to do the same thing. In fact I did just that for another family members pc a couple years ago. Ddr3 was at an all time low back then (had I known it would go back up in price I would have bought some for my future upgrade) and it was cheaper given those ridiculously low ddr3 prices to buy that family member a new motherboard AND the ram than it was just for the ddr2 ram. Otherwise I wouldn't have bothered with a slightly upgraded lga775 motherboard. That's not the case now with the higher price of ddr3. Now it makes more sense to move forward with a new socket.