However complicated a circuit looks, its parts are joined in only two basic ways: one after another, or side by side. Learn what each does to voltage and current and you can reason about most of what you will ever build.
Series: one path
Parts in series are connected end to end, so there is only one route for current. The same current flows through every part. The supply voltage is shared out between them, and their resistances add up.
Two 1 kΩ resistors in series behave like a single 2 kΩ resistor. Break the chain anywhere and everything stops.
Parallel: several paths
Parts in parallel are connected across the same two points. Each one sees the full voltage. The current divides between the branches, and the total current is the sum of the branch currents.
Adding a branch gives current another route, so the combined resistance is always lower than the smallest resistor. Two 1 kΩ resistors in parallel behave like 500 Ω. For unequal values: 1 ÷ R = 1 ÷ R1 + 1 ÷ R2.
The same rules for batteries
Cells in series add their voltages: four 1.5 volt cells make 6 volts. Identical cells in parallel keep the same voltage and last longer.
Why it matters
A voltmeter goes in parallel with the part you are measuring. An ammeter goes in series. A current-limiting resistor goes in series with its LED. Almost every rule of thumb in electronics is one of these two arrangements, applied.
If you want the fundamentals in one place, with hand-drawn diagrams, start with Getting Started in Electronics, or browse all four books.