Yeah the way you do this really depends on when you want the performance.
If you want to see a big performance improvement in games right now then you need to buy a new PSU and GPU and be prepared to spend a lot of money. You'd want to buy something high-end like a GTX 980 because when you get around to replacing the board, CPU and RAM (which you will want to do!) your GPU may be older than the current generation available at the time, but it won't be too much slower than the current generation for reasons I'll explain later. If you bought a GTX 960 or equivalent tomorrow it's true that you'd see the improvement from the 6950, but when you upgrade the rest of your system in a year or two you may find that your GPU is being a bit of a bottleneck and actually you want something faster.
However if you think you can live with your system for a bit longer and you are going to be prepared to save then I'd wait and build a completely brand new system using all of the latest generation components.
It also really depends on your finances. You need to sit down and think about this: how much do I want to spend overall AND more importantly how am I going to get this money and how long will it take me to get it?
If you have $500 to spend now and want to see an immediate improvement it'd be daft to build a new $500 system because for the same money you could buy a GTX 970 or 980 and a new PSU, put those in your current system and have a gaming PC that outperforms any $500 system you could build new. But if you think you want to spend $1,000 or more on it and you think you can save up relatively quickly (maybe in a year) then I would wait and build a $1,000 system when the time comes. The graphics card generation performance changes with every generation. The GTX 680 of 2012 performs the same as the GTX 770 of 2013 and the GTX 780 Ti of 2013 performs about the same as the GTX 970 of 2014 or 2015. It's true for AMD too: the high-end 5870 of 2009 performs the same as the more mid-range 6870 of 2010. Do you see that the high-end cards of one generation are equivalent to the more mid-range cards of the next? So if you saved $1,000 over the course of a year or so then by the time you are looking to buy a completely new PC you will more than likely be able to buy a card with GTX 980 performance for less than the price of a brand new GTX 980 today, meaning that it wouldn't really matter if you didn't buy a GTX 980 today with the intention of 'also using it in a future build' because you'll be able to get that same performance (or better!) for less money in the future when you are building a new PC using the current generation components.
So it really depends if you want to see an immediate gain or if you're prepared to wait. It depends how much you want to spend. It depends how much you think you can save and also how long you think it will take you to have the money to spend on a PC.
Whilst you are saving you will need to be so rigourous and not spend money on 'little things here and there' - all (or the vast, vast majority) of your money will need to put aside to fund this. If you are going to build something brand new then you want to spend the most money you can lay your hands on so that you can build the best PC you can afford, or think of it as 'the PC that will satisfy you'. Take myself as an example. When I was 12 I started saving for my first PC and money just falls through my hands like a sieve - it did back then and it still does today, to some extent. But by saving hard and not buying things I eventually built my first PC and it was a fairly high-end machine for its time and I was very happy with it. Right now I really want to buy a car and have driving lessons, but there's also a new camera I'm interested in and other things too. But I have decided I want the car and the driving lessons so I am not spending my money on anything other than that. You will need to be focused and see the end. But if you think you may be tempted to spend a bit here and there or only want to spend about $500 then I would suggest getting a new PSU and GPU is the way to go instead.
Hopefully this is helpful insight for you. Without trying to sound like your father, I'm trying to help you and I want you to spend your money in the best way possible.