Choosing the wood to use for building a door frame is a decision that will have lasting effects; many frames stand for decades. Before choosing a wood type, you should take several factors into consideration, such as frame location, structure type and costs. Based on these considerations, you can build a door frame that's both pleasing to the eye and structurally sound.
The wood choice for an exterior frame can be difficult. It has to be strong, secure against unwanted entry and durable, able to withstand extreme weather conditions including multiple freeze-and-thaw cycles. Your frame also has to go well with the existing structure and match the door it's framing.
You can go with mahogany or teak when building your exterior frame. Both provide quality looks combined with the kind of strength and durability that will last decades. The quality of these materials do come at a high cost though, so if price is a consideration, white oak will do a sufficient job, and is easily stained any color you want.
When building an interior frame, you don't have to be as concerned with your wood withstanding harsh elements, so you can go more with looks than with strength. Your primary consideration should be choosing a wood that goes with your door of choice and with the room into which you're installing the door. Common choices for interior frames are adler, maple and pine, with pine being the most popular.
Your interior door-frame wood should be easily painted to fit with a variety of decors or able to take a stain well if you want a more traditional look.