That’s sort of a bassackwards way for a business to arrive at their company logo (if that’s what this is). Then again, companies often approach design projects from the wrong end. Anyway, a logo is typically part of larger visual identity and branding project, not something to be subbed out as an afterthought for a website — especially considering that it’s for, as you put it, “a pretty big client.” Is this really going to be used as the company’s logo or, maybe, just an graphic or logo-like identifier for a specific project? It doesn’t really matter to your questions, though. It just sounds a bit off.
PrintDriver referred you to the Graphic Artists Guild’s (GAG) handbook, which is a great reference. Another very good reference is the AIGA Standard Form of Agreement for Design Services, which is way longer and more involved than a contract often needs to be, but it’s a fantastic (and freely downloadable) reference.
You didn’t mention where you’re from, so I’ll assume the United States. Unless this person is hiring you as an employee (which doesn’t seem to be the case), you’re an independent contractor (a business) being paid to do a specific job — it’s not a work-for-hire situation. If you start doing this kind of thing regularly, you’ll need to look into operating as a real business, filing quarterly returns, keeping detailed records, writing off expenses, setting up an LLC and all kinds of other crazy things, but for a one-off project, just write up a brief contract, get an advance, do a good job, get paid, leave with everyone smiling, then report it next year on your income taxes.
What do you mean by allowing you to get credit? No one will ever prevent you from taking credit for the work you do unless there’s a non-disclosure agreement involved. You might write into the contract, however, that you retain the right to use the work in your portfolio for self-promotional purposes. I’ve not had many client’s balk at this.
As for retaining rights beyond that, like trying to keep editable files from the client, well, just don’t. A business owner would be stupid to allow anyone to maintain control over the core piece in their visual brand. Your contract should state that upon payment in full, all intellectual property rights (like copyrights) to work product (the logo) are transferred to the client. If this were a brochure or an illustration or, well, most anything other than a logo, retaining the original artwork might be OK (depends on the designer), but for a logo, um, no — it should be 100 percent theirs once you get paid.
Always bill clients with an invoice (even if they’ve already paid you). It’s standard procedure in all cases. For one thing, they’ll need it as proof of an expense for their tax records. As for being paid an hourly rate, if that’s the case, write that and the rate into the contract.