I use an IRC bouncer to stay logged in to the IRC servers. I connect to the bouncer, which replays anything I missed on the IRC server while I'm not logged in to the bouncer.
I wish there was a way to say, "show me everything from the current conversation alone". Maybe by tracking @replies or something.
I've also heard of Bip (http://bip.t1r.net/), but I've never used it.
I run znc on a small VPS instance that is always running, always up, so I don't have to worry about keeping my connection to an IRC server open. ZNC (the bouncer) keeps me logged in to the IRC server, and I connect to ZNC when I need to be in contact with others.