Okay, a number of things to address here...
The issue you had Andy was (I assume, but a very well educated assumption) lay with the yahoo SMTP (Simple Mail Transfer Protocol) server used by your various clients to send email. The reason webmail was affected, as Roger pointed out but didn't get quite right) is that your personal devices have a public facing SMTP server specified which your client contacts when it wants to send an email. When you are using webmail the webmail client will be using an internal or maybe just different SMTP server reserved for Yahoo webmail use.
Roger - your comment about using port 80 is (almost certainly) wrong. Andy would be connecting to the webmail site on port 80 (http) or 443 (https) so everything he writes in an email uses those protocols to get the signals from his client to the webmail server but the actual processing of sending that email will happen over port 25 (SMTP) at the back end just to different servers (presumably unaffected by this specific outage).
POP vs IMAP - very simplistically POP is one way communication (server down to client) and IMAP is 2 way. Basically with a POP service emails are sent down to your client and either deleted from the server or a copy left there depending on your choice of setting. If you delete an email from your local machine then that email is still stored on the server.
With IMAP your client talks back to the server so if you delete an email from your client it will (if you have chosen to set it up that way) also be deleted from the server. This is, from an Enterprise perspective at least, a much better solution because it means that you don't have people's mailboxes filling up with old emails that they think they deleted (which they did from their client but it stayed on the server).
WRT getting all your emails over to a new service, there are a number of tools that can do this for you that you will be able to google but my advice would be NOT to set a forwarder from the Yahoo account to your new one but to actually do what someone else suggested and gradually change the registered address for any subscription emails and then deal with emails from individuals or more sporadic senders as and when they come in so as to completely vacate that address. Then and only after you have done this to a degree where you are not receiving any emails to that account, would I put on a forwarder to catch any stragglers.
The reason for this is that should Yahoo suffer another outage or even decommission their service then you have a lot more work to do in one go and risk losing emails.
HTH, any questions please shout. I'm by no means an email SME but I know enough and have a team of people who are SME's who I can ask if need be
Cheers
Mark