I have used my Polar Heart Rate (Smartedge model) monitor every workout for the last 3 years. I just replaced the battery on the watch, but the chest strap battery is supposed to last 5-6 years.
I think the chest strap is the better option because it offers continuous heart rate readouts. The ones without a chest strap usually require you to push a button or some other motion when you want to see your heart rate. Kinda of annoying in the middle of an aerobics class, huh? It is much easier to simply look at your "watch".
The strap simply feels like a stiff bra band. It actually took me more time to get uses to working out with a watch on my wrist than getting used to the feel of the chest strap. If you are used wearing a bra, you will not even notice it (men, who usually have never worn a bra find the chest strap takes some getting used to.) I usually put mine on under my sports bra and it does not show much.
Which brand / model you get depends on where and what you want to do with it. The basic , inexpensive models do nothing but display your heart rate. The more expensive models have a stop watch, calculate calories burned, and all kinds of other stuff. I would suggest getting model that has an alert when you are under / over your target heart rate zone. Since the whole purpose of wearing a monitor it to see if you are training in your zone, this feature makes the most sense. How many bells and whistles you want really depends on your budget. I think the prices range from $40 - $200+. I use the target zone and stop watch functions every workout and rarely use any other functions.
If you will be working out in a gym that has cardio equipment, you will probably want to get the brand that works with your gyms equipment. Many treadmills, bicycles, elliptical trainers, etc have an interface that shows your heart rate on the control panel if you have on a compatible chest strap. Most equipment is "polar compatible" but some works with another brand. Just ask the staff or look at the equipment.
About batteries, the battery on the watch should be replaceable by a jeweler. But the chest strap will probably have to be sent back in to the manufacturer to replace the batteries. One brand does make a chest strap that you can replace the battery in yourself, but friends who had one found that the battery compartment was not sealed tight enough and sweat got into the compartment causing a malfunction.
Generally, Polar is the dominant brand of heart rate monitors.