The axiom in network security is that the only truly safe and secure computer is one that is powered down, disconnected from everything, no one can see or touch it, and inside a Faraday Cage.
That may seem a touch extreme, but it's intended to illustrate the precarious issue of defining "what is secure."
First of all, you don't state what you happens after your computer connects to the wireless network. It's one thing to just "connect" to a wireless network. It's quite another to do anything across that network. What you do makes a huge difference.
It's like asking, "What can someone learn about me if I connect my driveway to the road?" versus "What can someone learn about me if I they see me driving on that road?"
Secondly, it's broadly inaccurate to ever assume one has a "fully secured computer." There's a plethora of unknown, suspected, and even known but non-resolved vulnerabilities across the spectrum of operating systems and applications.
So, to return to my counter-example. It'd be like asking, "What can someone learn about me if I drive on the road?" versus "What can someone learn about me if they look inside the windows of my house?" versus "What can someone learn about if me if they look inside this window I didn't know I had?"
Third, you don't declare whether the network to which you are connected is your destination or whether it's a transit network to reach yet another network. (Are you getting out of your car to go shopping once connected? Or are you using it as a freeway to go somewhere else?"
Fourth, you don't say if you are encrypting (SSL/VPN) or using some kind of proxy or what the nature of your activity would be - if any. (Are you driving in disguise? Is your car wrapped inside a semi-truck? Are you driving to a friends' house first and switching cars? Are you driving a EcoSmart cart, a bulldozer, or a sports car?)
But to help answer your question, let's assume, for a moment, you have a "fully protected" computer and decide to go to some resource on a remote network after connecting to it via a web browser. I'll also assume you use this wireless connection regularly and frequently:
They would likely know or be able to guess:
- The day/time of your activities and build a social/lifestyle schedule.
- Enough data to attempt cracking your encrypted logins/passwords.
- They would identify what websites and URL's you like to visit.
- Your OS and your browser type.
- Possibly read your browser cookies and history.
- Identify your social media (Facebook, Twitter, Instagram, etc) preferences and possibly your identity.
- Shopping habits, purchase preferences, age, gender, and lifestyle.
In short, anytime you connect to any network, one should assume the network has the ability to monitor any and all traffic you send or receive.