Microcontrollers are effectively a 'computer' on a single chip that contains a CPU, memory and programmable inputs/outputs (1). Consequentially, this provides an economical solution in the DIY situation to relatively easily digitally control a wide variety of devices and produce solutions to myriad problems, that a more traditional 'analog electronics' approach would require extensive electrical engineering knowledge and skill.
Various microcontroller families (PIC, ARM, AVR, etc) typically have onboard functionality such as analog-to-digital converters (ADC), Pulse Width Modulation (PWM), programmable timers, voltage comparators and general purpose input/output pins (GPIO) which with programming via EEPROM and flash enable sophisticated solutions with only relatively simple external ancillary circuitry. Such minimum external circuitry typically involves a suitable power supply, an oscillator and connections for the programming of the microcontroller.
Having choosen PIC microcontrollers (there are many sites comparing and contrasting the various microcontroller choices, I choose PIC mainly because of the extensive literature available) the next step is determining how to incorporate these components into a 'solution' (e.g. pump controller etc). A possibly easy option is to purchase a 'development board'. Such development boards include necessary minimum ancillary circuitry (power supply, programming connections, etc) and a variety of other typically required components (LCD screen, buttons etc).
The 'development board' approach (or even going with a variety of 'Ardinuo') makes it relatively easy to start using microcontrollers to accomplish 'electronic projects', however, in my opinion inhibits 'electronics' learning. Learning how to 'get a PIC working' from scratch is a valuable learning experience in itself, and since a wide variety of projects will only require a simple microcontroller circuit with the minimum support circuitry, enables DIY projects with the minimum components (i.e. in-expensive) and or custom made to whatever your particular requirements.
The minimum requirements for a PIC (PIC16 and PIC18 dealt with here) include a power supply, oscillator and programming connection (in-circuit serial programming ICSP). Power supply is easily provided by a simple linear voltage regulator (e.g. LM7805 supplied from a 'wall wart' or batteries). The oscillator depends upon the particular PIC, however, typically can be a simple RC network, a 'crystal' or an on-board internal oscillator, depending upon your circuit requirements. In-circuit serial programming (ICSP) is a straight-forward connection scheme/standard for programming PIC microcontrollers (requires a suitable 'programmer' e.g. PICkit2).
This minimum circuitry is demonstrated for a PIC (PIC18F248) with CCS C/PICkit2, using what I consider the simplest test application, flashing a LED (the electronics/microcontroller circuit analogous to the 'hello world' application of the 'programming world').