I've been looking throughout the SRD and I haven't been able to find answers for these questions. I'm sure I'm just missing them, but I'm not assuming I just know the answers from 1e.
How is Initiative calculated?
If you're Trained in a skill, is it a +2 or +3 bonus?
What are all the effects from being Small?
Initiative uses perception as a default. You can also use various skills if the situation makes sense. The most common one is using stealth if you were being stealthy when the combat starts.
Trained is 2 + your level. Don't forget to add your level.
And for reference: untrained is 0. Expert is 4 + your level. Master is 6 + your level and legendary is 8 + your level.
Smal has no inherent effects. It mostly limits what you can affect with combat maneuvers. Those usually only work against creatures up to ine size larger. For the same reason being small also makes you a viable target for maneuvers by more different creatures.
I'm not assuming I just know the answers from 1e.
Good plan.
How is Initiative calculated?
Initiative is the only opposed rolls left in the game. Everything else will be a roll vs a DC of some variety.
By default, your initiative bonus is your Perception. If you are doing something during exploration before combat starts (such as sneaking around, or negotiating with potential enemies) either you or the GM can negotiate a different skill check for initiative that is appropriate to the activity you were doing (such as Stealth or Diplomacy).
If you're Trained in a skill, is it a +2 or +3 bonus?
Trained is +2 +level. So a total of +3 at level 1, but that goes up as you gain levels.
You can also increase your proficiency rating which adds another +2 for each bump. So Expert is +4 +level for example.
What are all the effects from being Small?
Not much. It no longer changes your weapon damage or attack bonus or pretty much any other stats. There are some size difference limits to things like grapple and trip - so a small character won't be able to grapple as large of enemies as a medium size character can.
If you get beyond size small or medium there are some differences in reach, carrying capacity, and equipment weight.