Build, but try to either find a friend who has done it before, or look online for good, DETAILED, guides on how to do it.Not sure if I just buy a new desktop or build it.
(this is of course only if you have not done it before).
Also, when choosing pieces, remember to check sizes (that everything has enough room in the casing) and fits (like correct sockets).
There are many tools online to check this.
Also, be savvy. Try hardest to avoid any bottlenecks (no reason to buy an expensive graphics card if your CPU is not fast enough, do not buy more RAM than your software can use). Also do not cheap out on things like the CPU fan or the power source. Easy things to not care about, but can cause problems in the future if they are of worse quality. Lots of other things too, but check the guides.
You have to find the correct balance here. I have been part of building three computers myself, and every time I have done more work myself in building it. Last time (my cousin´s) I built like 90%, my older bro just helped check I did everything correct and helped me with the wiring).
It is a whole lot of fun, and the sense of accomplishment is high. You just have to understand that you bear all the risks. If you screw up, most pieces might be toast and $1000 bucks can disappear just like that.