I just started working at a small company (~250 employees) two months ago. I already have weekly 1-1s with my manager and team stand-ups, but I don't really have a recurring meeting with anyone else. If I do request one,
When first joining a new company or team, there's always a sense of excitement to do some variation of "making your mark, be visible, get an early win, etc.". Before I directly answer your question, I'd explore how much is driven by the urge I described, making you feel bad if you are not connecting with everyone on a regular basis.
Now with that out of the way, I'm actually a big advocate of using your "new person card" to get intro meetings with those within your team, department, org as well as the adjacent counterparts that you'll interact with for starters. There is content on Taro regarding accelerated new joiner onboarding, so certainly check that out.
Here are some suggestions to go about this more strategically to avoid the concerns you brought up and build lasting relationships (there' s a Taro guide on that as well actually).
Don't overthink it. Go in and have a normal conversation. Be genuinely curious about the other person and offer to share some things about you. At the end of the conversation, you should trust your initial gut-feeling on if a follow up meeting is need and at what frequency.
A piece of parting advice that is a bit philosophical. My heuristic is asking whether I feel better off or worse off after interacting with them. Better vs. worse doesn't necessarily mean happy vs. not, but do you feel like a slightly better version of what you want to be.
I would be start with a select few people for a recurring 1:1:
The first 2 people above (and probably the TL as well) have a vested interest in helping you onboard, so you should absolutely expect (and schedule) time with them for the first few months. The phrasing for these could be something like: "As I ramp up, I'm really eager to hear your feedback and learn from you."
It helps to have an end date (e.g. 3 months after you join), so it feels less daunting to get the calendar invite.
For the broader team, I'd start with just a single 30-minute meeting. Since you're new, you're still developing relationships and figuring out what you'll work on. It doesn't make sense to book a recurring meeting until you have more clarity on that.
Be proactive about scheduling the first meeting rather than asking for permission. Something like:
I'm new to the team and your name came up as the owner for this. [I saw you've been working on X.] I'd love to learn more about that and also hear about your experience on the team so far. I went ahead and found 30 min on your calendar for next week, but LMK if that doesn't work and I'm happy to find another time!
Regarding the purpose of the meeting, it can literally just be a "get-to-know-you" call. You can play that card pretty easily as the new person.