What are the pros and cons of collecting parts yourself, versus getting a DIY kit and then modifying it? I'm thinking about buliding my own 3D printer from scratch. 

Is it better to buy a starter DIY kit and try to build your printer around it, or to order separate parts for printer, and then to combine a printer?
From a general point of view, there are a few things to consider.

If you buy a kit:

Pros:


You get some insurance that you have all the parts that you need to get a functional printer - all the electronics, structure, bolts, nuts, screws, washers, wires and so on.
Most likely, all the parts you get are made to fit together.
You will (usually) get a manual, often a community that can help you out, and sometimes even technical support.
Sometimes, it can be cheaper than buying each part separately (but it can also be more expensive)


Cons:


You have limited/no options to customize your printer to your own preferences without purchasing additional parts. 
Some kits can be difficult to upgrade later or may be locked to some configuration or software.


My opinion:

The way I look at it, the better option for you depends on how you want to spend your time. That is:


If you get a kit, you can spend more time building.
If you collect all the parts yourself, you will have to spend time planning, ordering parts (possibly multiple times) in addition to actually building the printer. A possible lack of manuals could also increase the building difficulty.


If you don't already own a 3D printer, I would recommend getting a kit, simply because struggling with trivial things like parts not fitting together can take away the fun for many people.