The simplest almost-no-tool solution is to buy metal angle brackets. Without knowing dimensions, lumber thickness, type of lumber, or what tools you have, it's hard to give detailed advice, but if you put the horizontal piece on top of the vertical piece and attach them with angle brackets, it will be moderately strong. If you have other tools, there may be a better solution.
Apologies, didn't realise that would be useful. Like I said, woodworking moron over here
At the moment I have no tools nor idea what lumber I'm using (still researching plus I need to see what I can actually get hold of) but I'm willing to buy stuff. I don't even have a decent drill at the moment but I know I'll need one. I'll probably get the lumber merchant to cut all the stuff for me as I don't have a workshop (nor the room for one).
Here's a top down view with dimensions and a bottom-up view of the same, just showing the layout of the legs etc.
I don't even know what wood to use so I'm not sure if that makes a difference. I'd like a seamless join but I know that probably won't be possible - it will be sanded, and stained probably. I could always use a little wood filler, sand it down, fill again and sand just to fill the gap once I have a stable join, right?
It's probably worth noting quite a lot of weight will be on this. Probably 3 22" flatscreens (I'm a student studying computing at uni so you can never have enough screen space, right?) and a laptop plus various other bits and pieces. I need to design a little printer shelf too but I don't know how to attach that to the existing design. Either way I will be using reasonably expensive wood because I want this to last a long time so screwing up isn't really an option.
I didn't realise I could attach my .skp file so I've attached that too.
Edit: I read your post and realised you were talking about legs.... I mean the actual top part of the desk. Apologies.