I have currently contemplating transferring my domain to a new one with dreamhost. Is this possible? Will it cost me anything extra? Can I get more information about this?
Definitely yes. You can transfer domain to Dreamhost. You can refer to this article on how to do that.
Please note that is going to transfer registration to Dreamhost. After it is done, dreamhost is your registrar. You can another option to transfer only your site to DH but leave registration with your original registrar. If you choose this option, you can refer this article on how to transfer a site to DH.
will it mess up my billing cycle with dreamhost since my current host is dreamhost?
That may be true, if you have a hosting plan at DreamHost and you have not used your “free” domain that come with each hosting plan.
If you do not have a hosting plan, or if you have already used your “free” registration, you will have the additional cost of $9.95 for a one year renewal (with the year added to the amount of time left on your existing registration).
Nope. Registrations are to be paid when they are due irrespective of your billing cycle.