I've been trying to teach myself electronics. I've got some good textbooks and other materials but I'm still having trouble picking up a schematic and understanding it. I know the symbols and some of the theory. In order to understand what a schematic does, do I need to calculate all the voltages and currents between all the nodes, etc.?

 

No, you don't usually have to do any calculations.
Learning to read a schematic is somewhat like learning a new (hieroglyphic type) language. The more you read it the easier it becomes.
Usually the signal flow is left to right (inputs on left, outputs on right).
The power to the circuit usually flows from top (positive) to bottom (ground or negative).

When looking at a schematic try to follow the signal and determine what each element does in the chain as it moves through the circuit.
Often there will be stages with each stage performing a distinct function (single voltage amplification, signal filtering, etc.).

It will help if you become familiar with typical simple circuits such as the various types of BJT amplifier stages [common-base, common-collector (emitter follower) and common emitter] and the MOSFET amplifier stages [common-gate, common-drain (source-follower) and common source).
Also learn the differences between the terminal bias polarities for NPNs and PNPs, along with N-MOSFETs and P-MOSFETs.

Another thing to learn is various simple op amp circuits such as inverting circuits, summing circuits, non-inverting circuit with gain, follower circuit, integrator, differentiator, etc.

Understanding the virtual ground concept when using negative feedback in op amps is important.
Misunderstanding of that trips up a lot of people.

If you post a circuit schematic that you are puzzled about, someone here can likely help you.