Wanting to buy used car, how to negotiate the price?

My husband and I are college students and I only have one year left. We have a Honda 2005, that seems like there's something always wrong with it. Before my husband and I got married, he bought the Honda, but his parents told him he could only get that car or they wouldn't co-sign. The car still has 3500 on its loan and probably another two years of payments. It also has 235,968 miles on it.
We just took it for an oil change yesterday and supposedly the transmission is leaking as well as the axel, but we don't see any leaks.
Yesterday, we went to look at used cars and found a Hyundai Elantra 2013 for $13,400 and it has 51,000 miles on it. We want to get a better car that will actually last us, instead of fall apart on us and we do not want to put more money out on it than it is actually worth. I know this is a rough situation and the car is probably not worth $3500. I mean it is probably in fair to good condition.
How can we negotiate for a better price than $13,400?

I'm questioning the logic of again going into debt. You are already in debt on the Honda. The newer purchase will compound that problem. Your best bet is to pay cash.

Clark Howard, who appears on CNN, has tips and techniques for car buying on his site.

Doesn't matter-upside down.

You don't get out of debt by borrowing more.

Most dealers price their stock very close to fair market value. Modern buyers are savvy enough that pricing games are a thing of the past with main stream dealers. They may build in a bit of bargaining room so that you can think that you made a "deal" if you pay less than their asking price, but at your price point that is probably no more than $500 - $750 tops.

The real problem that you have is the car that you are driving. At best it might be worth $1,500 at trade-in and a dealer offering that much is not likely to budge on the price of the vehicle that you are trying to buy. That leaves you $2,000 short to pay off the loan. You either need that in cash or you have to roll it into the loan for the new vehicle. It might be fairly easy to find room to "bury" a $2,000 shortfall on a new $35,000 car but on a $13,000 used car that might not happen. And if it does, you'll likely be looking at a higher interest rate to pull it all together.

You're looking at $428 - $452 per month on a 4 year note depending upon the interest rate that you can get if you can roll the shortfall into the new note. I used 15% and 18% for those numbers. How does that stack up against what you are paying on the old car plus average monthly repair costs?

If you're Really lucky you can find some poor schmuck who will pay you enough to pay off the old clunker. That should reduce your payments to the $372 to $398 region and if you can get the APR down to 12%, down to around $352.

First step is to go to your bank or credit union and see where you sit. What are they willing to lend you? And at what APR? They can give you some ballpark numbers without pulling a hard credit inquiry or a firm quote with a hard inquiry. They can also give you the same Black Book numbers that the dealers use to price their stock. Then you have a basis for comparison and can then shop the best possible deal.