RC Servo control requires no special high-current capacity. These servos are controlled with logic-level pulses, so all you need is a GPIO pin and the ability to produce consistently timed pulses
This is pretty simple to do using the Arduino Servo Library. Learn how in Arduino Lesson 14
in the Learning System.
Even tiny processors are capable of servo control! Check out Trinket and Gemma Servo Guide
in the Learning System too.
The versatile Adafruit Motor Shield V2
is another option for servo control. It too uses the Arduino Servo Library, but provides 2 convenient pin headers so you can just plug right in.
The Motor Shield Guide has a page devoted to Servo Control here:
If you need to control more servos and/or free up some Arduino resources, you can use a dedicated servo controller, such as the Adafruit 16 channel/12 bit PWM Servo Breakout
. These are addressable, so you can control hundreds of servos from a single processor.
Or the Shield Version
of this contoller. It uses the same PWM chip and is addressable and stackable to control hundreds of servos.