I use Thunderbird to access my Yahoo, Gmail, and POP3 accounts from two computers (work and home.) Thunderbird can't, by itself, access the mailboxes of most free webmail services, but there's an addon for that: Webmail Addon
Thunderbird can be configured to dump all incoming mail into a single inbox (my preference) or each account can have its own inbox. Alternatively, you can set it up to have one inbox with subfolders and create rules that route mail to a particular subfolder based on sender, recipient, keywords, etc. Pretty standard stuff for an e-mail client. For example, I have mail from my mom directed into the "Family" subfolder, whereas mail from Ebay goes into my "Commerce" folder:
Thunderbird is rather more secure than Outlook or it's offspring, in my opinion. Whereas Outlook uses the Internet Explorer rendering engine to display e-mails, Thunderbird uses the same rendering engine as Firefox. While Neither Firefox nor Thunderbird are bulletproof, they do have a better security record than IE.
Syncing is not something I really bother about much. Yeah, I have two computers checking the same accounts, but never at the same time. I have noticed, however, that my POP3 accounts tend to honor the "delete" command whereas Yahoo and Gmail tend not to. It's probably a setting somewhere in the options dialog, but I'm too lazy to look right now.
Tracking and the like go hand-in-hand with using free e-mail services. While I don't necessarily believe that Google et al are using my info for nefarious purposes, I am cognizant that I am using their servers for free to send my communications. Thus, I never use them for anything truly sensitive. For such communiques I will use one of my paid POP3 accounts, and encrypt the message before even setting it loose on the internet. (Thunderbird can support GPG signing and encryption through the Enigmail addon
(GPG has to be installed already, though.)