There are three main ways to engage with individual users on LinkedIn: Messaging, recommendations, and introductions. Messaging is simple and straightforward. It is a way to communicate with your connections in a private setting. General messaging with connections is allow for all users, but InMail is reserved for paying members. It is important to remember to respond to messages as promptly as you can. People appreciate it, and it will help build your reputation on social media.
Recommendations are essential to successfully using LinkedIn. In fact, LinkedIn does not consider your profile to be complete until you have begun to receive them. You can ask people to recommend you for certain positions you have up on your profile, and they can also recommend you freely. Many people feel uncomfortable asking for recommendations (especially introverts), so consider recommending others to encourage them to return the favor. When seeking or writing a recommendation, avoid fluff, and go for specifics. You want to give your audience as clear of a picture as you can about the job done and the skills the person displays.
Introductions are also extremely valuable. If two people in your network want to connect, you may be asked to introduce them. It is not a complicated process, but people would be very grateful if you take the time to do a good job. You can elect to simply forward the request you receive, or you can think about ways to help the two people really connect and be successful. The latter will be far more appreciated, and this can only help you in the long run.
Take care to remember LinkedIn is not a numbers game. People often manage to build impressive numbers and engage and value each connection, but each individual has to decide what works for them. If you are a fellow introvert who might be overwhelmed by large numbers, stick to smaller, quality networks.