Why do my new basement lights trip the breaker?
Am I right in thinking that its the new pocket light tripping the breaker somehow, or could it be the cable, or is it still too many items on that run?


Probably none of those.

You have probably miswired a switch. You might have miswired a wiring box. This is easy to do since a white wire is not always neutral, particularly where there is a switch-loop.


  I then upgraded that breaker to a two 20 amp breaker


That is a good way to burn your house down without solving your initial problem. 

It is extremely unlikely that your light fitting draws more than 15 amps - that would be an unbearably dazzling and roasting 1800 watt fixture using at least 18 of the brightest and least efficient old-fashioned 100W incandescent tungsten-filament bulbs you could normally find. Maybe fewer if you are cooking your dining table under a large bank of those halogen floodlights designed for exterior use. 

It is dangerous to respond to a short-circuit by installing equipment that allows even more current to flow and allow the wiring inside your walls to run far hotter than designed for.

It may be that your circuit wiring is sized for 20A (but why?) in which case my alarm is unjustified (but if so, you should say so in the question).


  I am a novice DIYer.


Trace out the wiring, take clear well-lit well-focussed photos of it all and draw a wiring diagram. Figure out how it is actually connected. You'll likely find the problem that way. If not, post the photos and diagrams here.

Useful references


Electric shock - was I stupid, unlucky, or a combination of both?
When doing electrical work, what do I use to check wires are safe?