Hi Amira!
If you’re looking into an animal research lab, I would suggest reaching out no more than 2 months ahead of time. So, if you want to join fall semester of the next academic year, e-mailing in mid-July should be fine. This is for two reasons: a) there isn’t a ton of need to e-mail super far in advance, and b) it’s easier on PIs (principal investigators) and grad students if they meet you in person soon after you reach out. In the case of animal labs, you will have to complete ULAM animal training, but the online modules don’t take too long and the required in-person handling and training is scheduled much more frequently once the school year begins; they rarely hold it over the summer. 
However, if you are thinking of joining a clinical lab where there’s the possibility of interacting with human subjects, I would suggest reaching out maybe 3 months earlier than your planned start date. It’s very possible the lab could get back to you quickly, but there’s many more bureaucratic hoops to jump through in human research such as IRB approvals and HIPAA training, and these could take awhile (I once had to wait 2 months to get approved by an IRB to be on a study!). 
If you have any more questions about research, feel free to ask another question and/or check out our blog posts where we talk about how to find the right research lab for you, the types of research labs out there, and a sample e-mail (all the way at the bottom!)
Good luck!
Anni (Peer Advisor)