My favorite answer here: it depends. As Phil said, each company handles things differently.
As a thumbnail guide, most of the job ads I've seen treat 1-2 years experience as a junior role, where there's likely to be a fair amouont of guidance needed. 3-5 is usually seen as an intermediate level, where the tester is largely self-guided and is starting to mentor juniors. 5-8-ish is usually the senior level where a tester is expected to mentor others, act as a resource for the team, and often as a stand-in for the lead/manager. I've seen lead positions advertised at anything from 7-12 years experience, usually asking for technical lead type duties (mentoring, guiding the team direction, sometimes giving team members reviews, but rarely including hire/fire or budgeting). Actual test team management (as opposed to a title with regular tester duties) seems to be a 10+ experience thing - and if you want to continute to test, may not be what you want to be. Some places will explicitly designate technical leads who are "manager level" but choose not to follow a management track.
If you're looking for promotions, I'd suggest taking a good look at the duties of the next level you're aiming for, and start doing them as much as possible. It's pretty common for places to have an unspoken promotions policy of "we promote them when they're doing the work of the role we're promoting them to".