Start with the informal
Hopefully as a new employee, you have regular meetups with your manager where you check in, make sure you're getting feedback, and figuring out your next assignments. That's a good time to ask, casually "hey, are there mentoring programs here? how does one go about getting a mentor?" If your manager seems confused, or surprised, you can pretty much repeat the second paragraph in your question here - it's perfectly eloquent for what you are looking for and from whom and why.
That's going to be your first, best path - and it raises your manager's awareness.
Next...
Your manager may have some guidance you can go with. For example, he may want you to get through onboarding before engaging in a longer-term mentorship, so you get the short term, ramp-up focused advice from a specific source (who may not be the best long-term mentor). If you get thoughts from your boss - go with that.
If not... no worries. Many companies of smaller size don't think much about formal mentorships or have any structure for it... it just sort of happens.
If you get a vague answer with no real actions or clarity from your boss, take a week and follow up the next meetup with "hey, it sounds like we don't have anything formal. Mind if I set up a mentorship for myself?"
At that point, he may direct you down the formal route, or he may just say "fine, do whatever you like", but at least you get it clear that this is something you're going for.
Mentor Shopping
I maintain that the truly successful mentor/mentee relationships are the ones where on some level the mentee has chosen the mentor. Formal programs will help do the matchmaking and the reachout in places where there may be organizational impediments to having informal relationships. But the formal programs are never the only way to get a mentor.
The unspoken secret is that many people will just walk up to a seemingly wise person and ask. I've been asked several times... sometimes I've said yes, sometimes I've redirected the person in a hopefully helpful way (when I didn't have time, or didn't think I had enough to offer), every time I've been flattered to be asked. What helps is:
know what you're looking to develop in yourself. It's OK to say "I'm looking for my next promotion" or "I want to learn how to run a project like the XYZ project that you just did". Keep in mind that if you're shooting way high relative to your current skill level, the potential mentor may find a helpful way of advising a better interim step.
know what time commitment you yourself are prepared to make - weekly? monthly? do you have schedule limitations or other logistical things to keep in mind?
be willing to do the legwork - the mentor may offer to book the time on his/her own calendar, but they may have to ask that you reach out when you need it, and maybe even that if you two are grabbing coffee or a meal that you reserve the space, or do other legwork to make it easier on the possibly more time-strapped mentor.
at least propose an agenda. The mentor may offer edits or alternatives, but it's nice to not start with a blank slate.
