If I want to start my own business and copyright my name and logo as my first step, is this worthwhile and, if so, what is the best way to go about doing it?

Copyright protects original textual or graphic creations, and comes into being with creation--while filing for copyright adds some protection, just be creating something new, you have a base level of protection. You cannot copyright in advance of creating the text or graphic, and you can only get copyright if it is fully original.

Trademark protects an image, saying, slogan, name, etc. which denotes the origin or source of goods (or services, in which case, technically, it is a "service mark," though for all purposes, it functions as and is treated as a trademark). Because it denotes the source of goods or services, you can't trademark in advance selling/providing/etc. goods or services--or rather, you can file for trademark up to 6 months in advance based on an intention to use the mark in commerce, but then have to prove you did in fact use it in commerce within the designated time period. Trademarks have to not be used by anyone else, have to be distinctive, cannot be too generic or descriptive, and otherwise have to serve to identify business A as source of goods/services B.

Therefore, to get copyright protection, you have to create--express in a tangible medium--an original text or graphic; to get trademark protection, you need a mark meeting the requirements of distinctiveness, not currently in use, etc. which you are using or are about to use in commerce.

As for how to file for copyright and/or trademark protection; go to the U.S. Patent and Trademark Office's (PTO) website and read the isntructions there. If you are confident you understand them and can follow them, you can file yourself on-line; otherwise, retain an intellectual property attorney and let him or her guide you.