redirecting all mail to Googlemail

I am trying to redirect all my email to my Googlemail account so I can
read it on my PC at work. I have set up a (simple) rule in Mail app
that says: if any of following conditions are met/every message/
redirect message to [my Googlemail address].
But it only works s****adically for some strange reason and I couldn't
see any mail at work today (Tues) since early Monday evening.
When I got home I checked my usual email account (I have a Mailbo x
address as they were my original ISP but aren't any longer) and that
was working fine. But little if anything had been forwarded to
Googlemail since yesterday evening, but after I had opened G'mail
stuff started coming into it (but not everything in my Mailbox account
was forwarded). Forwarding to G'mail seems to work fine at home (where
I don't need it) but not at work (where I do). Should add that in my
Mailbox account everything is marked as having been forwarded even
when it hasn't been. I find this very strange and confusing.
Maybe I should set up a Yahoo or Hotmail account and have all mail
forwarded to that, is this a quirk in G'mail possibly? Or is there
some other way of setting up one's own webmail (Mailbox's which was
one fine is now terrible).

Can anyone please help.

Many thanks.