Really low interest rates and cash rebates are incentives periodically offered by the manufacturer. You want to know about these before you show up at the dealer.
In addition to the manufacturer's website, you can check out
www.kbb.com (kelley blue book) and
www.edmunds.com to find out the MSRP and any incentives being offered for the car you are interested in. Ideally, you won't pay much more than the MSRP for the car, sometimes you can even get them to go below MSRP. Unfortunately, it doesn't look like Honda is offering any incentives on Civics.
My advice is:
1) Figure out what the best rate you would qualify for by visiting the e-loan website (don't submit an application). With your credit score, the dealer will almost certainly be able to match it (or do better if there is a low interest rate incentive being offered). I didn't have any proof with me of the e-loan rate, all I said was, "I can get such-and-such with e-loan." and the dealer matched it. FYI, Captial One is offering 5-6% loans on new cars right now.
2) Go to the kbb and edmunds websites, look up the car you are interested in, and print out the prices (including the various trims and options, the sites have some very nice matrix-type charts). Take this with you to the dealer, they will know you mean business.
3) On these websites, there is usually a link to "get a fast dealer quote." This is a good idea because it puts you in touch with the "internet sales manager." The ISM will know you are an informed customer and will skip most of the haggling. They also seem less "sharky" than the floor sales guys, IMHO.
4) Get quotes from at least 2, ideally 3 (or more?) dealers. Play them off each other, or just go with the lowest quote. You may need to email or call regularly, or you may need to visit in person a couple of times to get all of the details (price, model, options, etc) worked out.
5) Once you have settled on a vehicle and the final price, the ISM may pass you off to another person to seal the deal, and
pile on the extras. So, decide well in advance if you want an extended warranty, Lo-jack anti-theft system, etc, etc, etc. Otherwise, when all you're thinking is "What if something happens to my beautiful new car?" - you may get suckered into agreeing to thousands of dollars of extras, completely nullifying all the hard work you did to negotiate the price down.
6) On other thing, don't forget to arrange for car insurance in advance. Call around and get quotes on that, too. You may need a VIN number to get a quote, don't worry if it's not the exact car you ultimately end up buying, you just need it for the quote. Once you've settled on the exact car, tell them the real VIN number, and have them fax the ins card to the dealer so you can drive away with your new car.
Good luck!!!
P.S. As far as your downpayment, with your credit score, you probably won't need the DP to qualify for a good rate (but read the fine print on the eloan and Capital One sites to find out for sure). So if you get a good rate, then it is really up to you whether or not you need to apply the DP to lower your monthly payments to a comfortable level, or if you want to use it as an emergency fund, or invest it in a retirement fund - whatever you are most comfortable with. If you think you will blow it, then by all means apply it to the car purchase.