Any brower updated within the last few years is safe. However, more important is the protocal being used. HTTP can be sniffed and spyed on, so it is not safe for transactions. Whenever your about to type in your credit card info make sure your browsers address field starts with HTTPS. This means they are using a secure socket layer 128bit encryption and you'll be very safe. I make atleast 4-5 online purchases a month. Just make sure you buy from well known companies like amazon and you'll have a very pleasent online shopping experience
Even though you're using secure http (you might also want to check for the lock at the bottom right on IE) i don't beleive that it means your neccessarily using 128 bit SSL. It could be a weak older version. It's recomended that you do a windows update and install all the critical updates and IE service packs.
Also, even if you are using SSL that only protects you from evesdropping crackers. the information is deciphered back on the servers so the person running the site can look at your credit card number, etc. There is still a posibility that the owner might go rogue and steal your information, but as long as you go with the big websites (as greg said) you should be fine.