Don't get the kind that just straps to your wrist, you need a chest band. Accuracy isn't as big of a problem as not being able to see your heart rate any time you want, because most of the wrist models make you press and hold a button or two for a few seconds to get a reading. Plus this makes you lose out on seeing average heart rate, max heart rate, things like that. You won't have high/low alarms or anything.
Even the entry level polars do more than most people want. But, when you get into the nicer ones... I'd really look at the GPS enabled ones from Garmin. I have the Polar 625x and there's a few things that are disappointing.
First of all, most people will have to buy an additional $50 infrared adapter to make it work with their computer, while the Garmin has a simple built-in USB port.
Also sometimes the foot pod for running just isn't accurate, sometimes it's spot on, but other times it's as much as 10% off or so, like if I know that a run is exactly 4.3 miles, it'll say 4.8. That's not THAT big of a deal, but this screws up your pace as well. If you enter a 5k race and want to keep a certain pace, it will tell you that you're going faster than what you really are, sometimes 30-45 seconds per minute faster, which is a pretty big deal. Garmin on the other hand is GPS based and even if you ran 100 miles, the most it could possibly be off is a few feet, but the polar, crap, I wouldn't trust it to be within 5 miles of being accurate.
Also the polar's altitude sensor is based off of air pressure. So, for it to work accurately, you have to constantly reset it with the current altitude, which basically means that you have to consult a secondary GPS. Otherwise, during the course of a day, even if you are on flat ground, it can say that your elevation is changing. Plus with it being air pressure based, it's just not accurate enough to really take into consideration on hills or whatever. The Garmin, once again, is GPS based, so all that functionality is built in and automatic and is very accurate so you can see exactly the pitch and height of a particular hill.
On the bike, you have to have a secondary speed sensor for the polar which is another $45 or so. The Garmin is GPS based and doesn't need a sensor.
Last but not least, the Polar doesn't tell you how to get home if you get lost.