What do I say after someone accepts my LinkedIn connection request?

It depends on who they are. For a recruiter: thank them for connecting, mention one or two specific things you know about the company (show you've done research), briefly describe your relevant background, and ask if they'd be open to a conversation about your candidacy — or if there isn't a role right now, express that you'd love to stay on their radar. Keep it short and specific. For a VP or hiring manager: similar approach, but also name a specific role you've found that you think you're aligned with. The ask is not "can you get me a job" — it's "I'd love to have a conversation" or "I'd appreciate it if you could forward my information to the right person." Always make the ask explicit, and always make it easy for them to say yes.