When your listening to them, take on board what interests them or where they are facing challenges, then:
- Send them links to articles they may find useful/interesting
- Just drop them a friendly/polite note in a week or two's time, giving them a super quick update on what you've been up to and seeing how they're getting along (try to reference something specific they previously mentioned e.g. how did that important meeting with investor A go?
- If you're starting to move in the same circles, you may bump into them at events, or you can specifically tell them you're going to one if you think it would be of interest
- Hopefully they'll also be in the social media circle too, so they will see content you produce on LinkedIn, Twitter, FB, HN, Quora, your blog etc.
- Finally, ask them if they want another meeting 4-6 weeks later...you'll have to gauge this on your first meeting. If you got on like a house on fire, then you can casually invite them for a drink and not worry too much about; if it was a little more formal, but you still think the relationship could be worthwhile for you both, be sure you have some interesting progress to talk about and have thought about value you can bring to them in the f/u meeting.
Networking and meetings a very much art over science I find, and you have to use a lot of personal judgement (and you won't always get it right) but if you can be yourself, add value and like the other person then these 3 ingredients should be just right for development some meaningful and lasting relationships.
Good luck with your meetings, let me know how they go!