I was thinking since it's almost New Years maybe a good idea to send an email to some of these people I'd like to maintain a connection with and wish them well. Maybe also cite a paper of mine ( one of them just got featured in a science magazine) , or would that be too 'showy'.
Unless there is a specific reason for you to contact any of these people, no I wouldn't do this. This isn't networking, it is a bit of a time suck for the people you're contacting. What sort of a response would you expect?
Think of it more as an exchange - whom do you like hanging out with at conferences? Who are your peers? Whom do you like playing (as in research) with? Whom do you talk to in your research/teaching life
Your peers (ie other postdocs) are as important in your networks as those you perceive as able to do something for you. What can you do for others?
networking and making connections, people that I can call on for letters of recommendation and who will speak well of me and my research
This isn't a network. This is a group of people whom you think will do you favours. What do you have to offer? How can you make a contribution to your discipline? A network is a give and take between people who enjoy working together. It's not hierarchical, so it's important to include your postdoc peers, or if you're still doing a PhD , other PhD candidates you meet in your area.
What are the major learned societies/scholarly associations in your discipline? Are there a couple of major conferences each year - could you attend one? Is there a postgrad caucus or ECR group or network? If not, can you volunteer to build one?
The way I think of networks is that they are professional connections which work sociably - so be sociable. Be someone who is there for others - then others will think of you. Suggest initiatives - if you want to be part of a network, start building it yourself.