There are several bits to this puzzle. You can "DIY" parts or all of it, or you can contract out parts or all of it.
First: the merchant account. Advantages are generally the smallest percentage overhead, over using something like PayPal. Another advantage is if you have multiple venues -- say, website, brick-n-morter shop, and occasional festivals or off-site charges -- this allows you that versatility. If your client already takes credit cards, this may be a given.
The opposite extreme would be PayPal. The purchaser doesn't even need a PayPal account in order for PayPal to "launder" the transaction through a credit card. This is by far the simplest on-line method: send them a PayPal invoice, and money appears in your PayPal-linked bank account. But it has a pretty steep transaction cost, and you can't use it anywhere that doesn't have an Internet connection. Ideal for someone who doesn't yet have a merchant account and isn't sure they need one.
If you go the merchant account route and take credit card info via a web form, it must be encrypted via the https: protocol. Then you must either "process and forget" or store it encrypted.
Email? Don't even go there. Chopping the credit card info into bits is a huge pain for both parties to the transaction, but only a minor problem for anyone who might be sniffing your email -- they can put it together just as easily as you can. The only exception is if your customer is sophisticated enough to encrypt the email and get you the key separately, in which case, they might as well just call with the credit card number.