My daughter is 28, so when she was pre-teen and teen the Internet was not so much the issue, but I see that others who have responded have addressed that issue nicely.
The lying was an issue though and it started right around 13. I was broken hearted at first when I realized that my child had lied to me. A good friend of mine who is also a psychologist said to me, this is not so much about you as it is about the fact that your daughter is becoming a teenager and part of her mission now is start testing her own wings and to learn how to break away from you, but you have got to let her know that you are still in controll because she needs to know you are strong enough to accompany her through these turbulent times she is entering. Part of what this means is that you need to prepare to be unpopular for the next 5 or so years. But you must stand your ground as a Mother who loves unconditionally and who knows something about life.
So what I did was to take away everything that she valued (TV from her room, music, excursions with friends, phone privileges, etc.) and told her she had to earn it all back. Then we drew up a contract to lay out when and how she could earn each item back. she had to exchange certain behaviors and "chores" along a timeline to earn her valuables back. Chores and behaviors included things like getting her grades up; keeping proper bedtime without me nagging; keeping her room tidy; walking, feeding, and washing the family dog on a regular basis; having regular days in each week to clean the kitchen after meals; etc. After two weeks of keeping her end of the contract she got her phone back, after one more week she got her music back, and so on.
I followed the contact to the letter and she did too. When she had all of her valuables back we wrote out a general contract where she had the potential to earn a certain weekly allowance in exchange for continuing the chores and behaviors. Each chore and behavior was given a monetary value for example 50 cents each time she cleaned the kitchen, 50 cent for each day in a week that her room was tidy, etc. (of course this was the 90s so you'll have to take inflation into account:)You what, this worked! The teen years were still tough but we now had a system in place for dealing with many of the issues that arose.
Also, I underscore what others have said in their responses about spending one-on-one time with your daughter. Set aside some time each week for the two of you to do something together--go to a cheap restaurant; go to a movie; go bowling; window shop at the mall; visit a video arcade; walk in a local park for an hour or so; Anything where the two of you get away from your regular concerns for an hour or so and just talk about unimportant stuff. This will make it easier for her to talk to you about important stuff when it comes up. This habit of regular Mom/Daugher time will also let her know that you're interested in who she is, even when there is stress between you.
Anyway, I've said enough. Just hold on to your daughter and know that you are in for a wild ride through the teen years, and take heart that holding her to standards and keeping limits around her will pay off in the long run. Our teens need need limits/boundaries and they need to know that we care enough about them to put up with a little pain (being unpopular, even disliked by them) to keep them safe.
My daughter has turned out great, AND she has since thanked me for the boundaries I set when she was a teen and for being strong enough to hold my ground with her.
Your daughter is lucky that you care as you do; and you are lucky to have her...