First off, where are you located and how far are you willing to travel?
If you've been up the Matterhorn (with a guide, I'm assuming), you now know what it's like to be on serious alpine terrain at high altitude. This is good experience to start, but you are probably going to want a bit more experience if you are willing to attempt bigger objectives in that realm, especially un-guided. Start on smaller peaks with less committing routes, and gradually work up to bigger and badder mountains. This takes TIME and patience. Unless you want to be guided up every big mountain that you will climb (maybe that's what you want....?), it works like any other activity; start off small and work your way up. Taking guided trips will certainly expose you to the hardships and skill it takes to climb extreme things, but you will learn so much more if you and a good partner spend the first few years of your climbing career tackling smaller objectives, and slowly working up to more and more technical summits.
In terms of specific things to climb, we need a bit more information.