I've finished my first novel, run it through a critique group, polished it up—done all the things you suggest we do before we submit. I think it's ready to go.
I want to try the national market. Where do I start submitting? Do I submit to publishers or to agents? If I go through an agent, what is the standard cut I should expect to pay them and will I have to cough up any money before the book sells?
Whether to submit to an agent or to a publisher depends upon your book (ie: which publishing companies are a good fit for you) and the publisher. Some publishers take unagented submissions. Others do not.
Lots of writers successfully sell their first book without an agent to represent them. They research publishers, find those that take unagented submissions, and go for it. Some of them get great contracts and healthy advances. Once they've sold that first book, it's a lot easier to get an agent to represent future books.
If you start with an agent, rather than a publisher, the process is basically the same. Research agents, find those that are accepting new clients, and go for it. The advantages of having an agent is that a good one will help you polish up your story to make it more sellable. They also have connections to publishers—they know who is looking for what type of book, who is good to work with, and agents can usually get better contracts and healthier advances because they (hopefully) know a few more things about the business end of selling books that newbie authors don't.
Most agents charge 15% commission on U.S. sales. Some of them require you to pay office expenses (copying, postage, etc.) before the sale, others deduct them from royalties (on top of the 15%). If they charge reading fees or an hourly rate, look somewhere else.