I received a simple electronics kit as a gift. It includes 80+ schematics but no explanations. I'm trying to calculate voltage, current, and resistance throughout each schematic but I hit a roadblock after only a few pages.

In the All About Circuits book, I'm up the DC Network Analysis sections. It's all straight forward enough for resisters, batteries, and other components with only 1 lead in and 1 out. However I can't figure out how to apply the rules with components not like that.
- Quad op amp. If only one of these is being used, I have 2 leads going in, 1 out, V-, and V+. If several are being used, I have the added complexity that they're sharing the V- and V+
- Transformer with 5 leads
- Transistors with 3 leads
- Logic gate (This one isn't in the kit, but while we're on the subject...)
- Diodes. Here you can't start by assuming either flow direction will work

How does the math work for these?

 

Quad op amp. If only one of these is being used, I have 2 leads going in, 1 out, V-, and V+. If several are being used, I have the added complexity that they're sharing the V- and V+

Well, you look at the power requirements for the op amp, which is given in the datasheet from the manufacturer(s). Then you ensure that the amplifier is given an adequate supply of current at the voltage it (or your circuit) requires. Op amps can be difficult to "get the hang of".
Here is a link to a really good tutorial on op amps:
http://cktse.eie.polyu.edu.hk/eie304/Op-amp.pdf
It will help you to understand how they work, and some of their limitations. The 741 op amp is dissected beginning page 24, with a discussion on it's limitations and how it could be improved upon.

- Transformer with 5 leads

The secondary is center-tapped. If the voltage from end-to-end is 12.6, the voltage from either end to the center will be 1/2 of that. Center-tapped transformers are very frequently used in power supplies.

- Transistors with 3 leads

Yep - if they only had two, they would be diodes!
They are current amplifiers. Basically (sic), a small amount of current on the base causes a porportionately larger flow of current between the collector and emitter.

- Logic gate (This one isn't in the kit, but while we're on the subject...)

Logic in a box You can do some amazing things with logic gates. Some of them are even quite handy for use in analog applications - see my "Simple PWM circuit" in the Projects Collection forum for an example of a CMOS logic gate being used in a combination analog/digital way.

- Diodes. Here you can't start by assuming either flow direction will work

Well, not if you're using a DMM set on Ohms. The meter has to supply a voltage in excess of the forward voltage drop of the diode; for silicon diodes this is generally between 0.6V and 0.7V. Germanium diodes (somewhat rare nowadays) may be half that much. Zener diodes are used differently than standard silicon diodes.

How does the math work for these?

Logic gates use Boolean algebra and Karnaugh maps.