I would like to purchase several domain names for a company I am
starting.  What is the correct way to go about doing this and making
sure I have reserved my domain names through the proper channels so
that no one else can purchase them??

I have purchased 3 domain names through 1and1.com.  Does this ensure
that they are mine? or do I have to buy them elsewhere too?

Thank-you for posting your question.

Once you have purchased your domain names through 1and1.com they
become your property for a fixed period and no-one else can register
these names during this period.  From what you have already stated you
have done this correctly and you will not have to buy them elsewhere.

You can check how long you have registered your domain name here:
http://whois.uberdose.com/whois/

Simply type in your domain name into the box and who the domain name
belongs to and for how long should be displayed.  If you have any
doubts about the answer that this gives you please ask for
clarification and supply one of the domain names you are querying.  I
will investigate the domain name for you and report my results here.

You have to register each domain name "extension" that you require. 
For instance if you wanted the name emaley-ga you could register
emaley-ga.com for universal coverage (most people guess .com first!). 
If you also wanted a presence in the UK would might also want to
register emaley-ga.co.uk to stop someone else "pretending" to be your
company.

1and1.com have a series of useful FAQ here:

Q: Once I register a domain, will I own it?
http://faq.1and1.com/domains/domains_explained/ordering_domain_names/2.html

Q: Can I change the name of a domain after it has been registered?
http://faq.1and1.com/domains/domains_explained/ordering_domain_names/3.html

Q: How many years in advance can I register a domain?
http://faq.1and1.com/domains/domains_explained/ordering_domain_names/5.html

ICANN, Internet Corporation for Assigned Names and Numbers, also has a
comprehensive list of frequently asked questions here:
http://www.icann.org/faq/

These FAQ should hopefully put your mind at rest.

If you require any further help on this subject please ask for
clarification and I will do my best to respond swiftly.