Clean the tips with pure acetone after you're done using them, or shortly before you're ready to start gluing. Cyanoacrylate (superglue) isn't water-solvent (that's why we can use it in our tanks), which makes cleaning it difficult. It does, however, dissolve in acetone. Let the tip soak in a shallow bath of the acetone; afterwards, the hardened plug has usually softened enough that you can work some kind of probe (a finishing nail, for example) into the plug and free a channel for the glue to flow through.
Alternately, you can find a superglue with a removable blunt syringe tip, like Maxspect's glue gun. Syringe tips are literally pennies each, so treat them as disposable. Use for one session, then discard afterwards and swap out with a fresh tip for your next glue session. That way, the glue at the actual cap of the tube doesn't have a chance to harden, and you can get more or less the full use from each tube.
If you do decide to give Maxspect's gun a try, give me a PM before the first time you use it. Their instructions are... less than perfect. The first time I tried to use it, it took me close to an hour to actually get it working properly. Once I figured it out, though... I don't think I can go back to regular tubes of glue, it's just so much easier.