I wouldn't consider myself an expert on LinkedIn by any means, but I recently revamped my profile and started trying to use it a bit more proactively, so I guess I can explain what I've learned. If you're familiar with LinkedIn some of this stuff may be obvious.

Your profile should look like your resume, but a bit more conversational. Full sentences instead of bullet points. Feel free to go into a little more detail bragging about skills and big projects if you have relevant stuff to mention. As far as skills, I'd take a look at anyone you know who does your job and compare.

LinkedIn visibility works by degrees of separation. So if I'm connected to Jon, I can mostly see everyone Jon is connected to, and they can see me. If you're in the same groups or follow the same tags/subjects, I believe you can see limited info about a person. Of course some people have their profiles more or less private than default.

Visibility in searches is based on profile completeness and number of connections, I believe. You can also set yourself as open to job opportunities for everyone, or open but only visible to recruiters. I've done recruiters only, just cause I don't want my current employer accidentally stumbling upon my profile.

LION - Linked In Open Networker, means they will basically accept any connection request, even if they don't know you. A good way to boost your numbers, but probably doesn't look great if you're only connected to them and not any "real" contacts.

When you add someone there's a 300 character limit, optional message. I added a bunch of random people from college and high school who mostly accepted my invite without a message, though.

You can't send messages to people you aren't connected to unless you pay for a pro version of LinkedIn. Paying for LinkedIn allows a whole bunch of extra stuff that I'm sure is useful, but I haven't tried it so can't comment on that.

As far as using LinkedIn as a tool to look for work (outside of the job postings), I recently had someone tell me to look up companies and go through the list of employees. They recommended messaging or connecting with recruiters/talent acquisition/HR, or people I had a tangential connection to (2nd degree connection, same university, whatever), and ask for advice on applying to the company. I've tried it a couple times, with no luck so far. But the basic format I was told is:

1. I found your profile while researching _company_.
2. I'm interested in company because ______.
3. In your experience, what is the best approach to apply to this company?
4. Thanks for your time/advice.

Hope this helps!