Hi Dough. I agree with Bugbatter tath eventhough all main webmails have some antivirus scanning mail, but you can not solely rely on them. Also it is not a matter of which browser or webmail is more secure or better. Remember any website or program may be compromised one time or another. However there are a few things you, as an user, can do to protect yourself. Like BB says above. It is your responsability since it is your PC, your account, and your identity that you want to protect. 1.) Keep your OS ( including your browsers ) and security programs up to date. There are programs like Java, Flash, Reader that are major targets by malware so that they can be used to infect your PC. If you can, avoid them, if not, keep automatic update enable so that any security hole can be fixed in time. 2.) Use a strong password. Combination of 10 or more letters, numbers, and symbols using capital letters too. Use different passwords for all your sites; mails, social, banking, and hange them regularly. 3.) Do not be click happy. Do not open unknown e-mails and if a friend sends attactments, call him/her or reply him and ask if he/she send those attactments. If something looks fishy, it probably is so do not click it. Do some research first. Have a special folder where you can download your attactments and scann them again. Do not open them directly fron the site, and if possible have a secundary scanner to have a second opinion. Malwarebytes is a good choice. 4.) In open wifi hotspots ( coffe shops, airports ) switch your firewall to " Public Mode " if possible, and avoid banking transactions on cibersites using a public PC. 5.) Have a backup strategy. Do regular backup of your system and important documents ( Files, pictures, videos, music, Fav, e-mails ). Win 7 and Win 8 have incorporated, I understand, a very good program to do that. There are other third party programs that can also do an image of your HDD regularly.  I think those are the mayor points to be relatively safe, but I imaging other members in this forum can add a few more.
I use Internet Explorer 10 & Google Chrome to surf the web. I feel they have the most to loose, if their browser has security problems However, that also makes big target. One Yahoo e-mail account was hacked into. How I don't know. Then spam was sent to all my contacts. I have used e-mail accounts, with all the well known web site programs besides Yahoo; Hotmail/Live, AOL, Gmail, mail.com and inbox.com. They all seem to work. With spam being a problem on Yahoo & Hotmail. To me your browser must protect users against viruses and malware, and other things of this nature. Because surfing the Internet is what you want to do with a browser. How are most of these things being spread? What is the most secure browser and e-mail provider. 
The most secure browser and e-mail provider & how are these viruses & malware 
problems being spread?