I have a website with a domain name that has been live for 2 years,
but now the business name is changing and there is a new domain name,
I am wondering what is the best way to continue marketing the website,
should I start to market the new domain name or should I continue to
market the old domain name even though the site structure consists of
the new domain name? or I should 301 redirect the old domain to the
new one?

I feel that this question is best answered after thinking about the following.

How strong is your initial brand name, why was the brand name changed
in the first place, if it was strong why was it changed? If it was
changed because it was weak is it worth continuing marketing a losing
brand name?

For example, changing from John's autos to Peter's autos is not going
to matter at all to anyone outside of your existing clientelle.

Changing from Toyota to "Chung Lee's Car Makers" is going to be one
hell of a shift in domain name if Toyota did not keep their old domain
live.

My advice for you is this,

On your old company domain, keep the site alive but have text saying.

"The company name has now changed, please use the following web address
(give new web address)."

Then set up a meta refresh so that after 5 or 10 seconds the user is
then automatically forwarded onto your new site.

This way both the OLD domain name is still advertising and helping the
business keep your current customers who know you by your old trading
name. As a bonus they will also learn of your new trading name.

New people who find your old information or who have had it given to
them won't hit a dead website and look elsewhere, they will be
informed of the new address and possibly tell your old clients of the
name change.

You now only need to market the new domain name as anyone hitting the
old domain will be old clients or have old information, anyone new
will be hitting the new domain. After 6 months to 1 year you will be
able to auto redirect the old domain to the new domain without needing
the change of domain name information on the site.

It may be wise to keep ownership of the domain name in case cyber
squatters try to steal it and ruin your domain.

In short, put most emphasis on the new domain but keep the old domain
live to catch any old customers, slowly phase everything over to the
new domain and after 6 months set up the old domain to send clients
directly to your new domain name.

I hope this answers your question, any other thoughts, leave me a
commment and I'll get back to you.

--Keystroke-ga.