What's your Strength Training routine?

Ive read that you want to build muscle because muscle burns more calories, boosting your metabolism.

Essentially this is correct. A higher muscle content, increases energy demand to drive it.

Then I've read opinions that you cant build muscle when trying to burn fat????

At the beginner/novice level it is easier to do both when you have a normal to high Body Fat %.
The lower your Body Fat% gets the harder it is to drive the body to release it. For most MFP purposes it IS possible and there are numerous people doing this including myself :-)

And, dont mix cardio and weight training on the same day. But then Ive seen workouts that include doing a set of weights, then cardio for 5-10 min, then another set of weights. Everything is so conflicting!

There are some concepts here that need to be understood.
In general if you do a lot of cardio before a weight training session, you will have partially fatigued those muscles in the cardio session. However it depends on the intensity and duration of the cardio session.
Do not confuse "a cardio session" with a recovery walk/run/xtrainer warm-up or in-between exercises recovery.
Between weightlifting sets commonly referred to as REST period, I will do brisk walking. this keeps my heart rate up and helps flush blood into and out of the muscles aiding recovery for the next set. The recovery intensity is light, but not a dawdle!! ;-)

Then there is the general term, "strength training" What exactly does it mean? Free weights? machines? body weight workouts? dumbells? High reps, low weights? High weights, low reps? Again, so much conflicting info. I read the term over and over but dont really know how to do it.

This is resistance training as opposed to cardio training which is time based, it is weight based; Weight on a cable, weight on a barbell or your body weight.
In general if you are after increasing your strength over muscle size, you will tend to be doing High Weight/Low Reps. Rather than go into the detail of this here, there are some basic weight training books available that cover these details with out the 'broscience', but are based on decades of use and understanding of how the human body adapts. The first is 'Starting Strength" by Mark Rippetoe and 'The New Rules of Lifting for Women' by Lou Schuler, Cassandra Forsythe, and Alwyn Cosgrove.
These are the best places to start.

Now how about your target zone heart rate? Is it really just a number from a chart based on your age? When do you want to be in what zone? For how long?

Here you are talking about Cardio, be it Running, Cycling, Swimming, or X-trainer. A lot of research has been done over the years and those tables are general guidelines based on the collected data. Another book that will hold you in good stead is 'Fitness for Geeks' by Bruce Perry.

To make sense of all this Im wondering what everyone here is doing and results you've gotten. Please be specific on your workouts.

JWoffindin abbove is correct in that we all have slightly different goals and training routines.

But for what its worth, here is what I am currently doing;

The first five months of this year I was focused on losing weight (fat) with out injury, and was walking/jogging three times per week and doing weight training (8-12 reps) 3 to 4 times a week. Some days I would mix the cardio with the weight training.
After losing about 20kg I was at a point where I was running good distance and pace and had started to plateau on some of the weight lifting. At this point I researched to change my program and get more fitness focus.

I chose to go with Strong Lifts 5x5 with accessory exercises. Here there are five main weight lifting exercises focused on increasing beginner/novice strength. Squats, Deadlifts, Overhead Press, Bench Press, and Bent-over Row.
Strength training three times per week only!

For my running I am now following a program generated using the www.micoach.com website and downloaded into my phones miCoach app. miCoach covers a number of activities running, football, etc but I use it just for running. I set a goal I want to achieve, the number of days I want to train per week and the date I would like to achieve the goal. The website prepares the training plan for me based on a lot of knowledge of a lot of coaches!!! :-) miCoach takes the complexity out of it but you can dive into it if you want.
So back to my cardio....
I do one to two long/easy runs per week, and two speed runs per week. these speed sessions are based on pace and heart rate in the ZONE. Zones are worked out individually using the app and website based off an assessment run.

So hows my progress, awesome!!! My strength gains are measurable and my run pace is improving at about 15s/km every 2-3 weeks.

Hope this is of some help to you and others. I wish you well :-)

weight training is whatever your workout is utilizing some type of weight. Could be machines or free weights.

I was going to respond but people posted some good stuff. My best piece of advice is to learn as much as you can about nutritn and exercise and keep up on it.. It is forever changing.

Dangermouse gave some good advice. I would definitely not wait to begin lifting. Unlike a previous poster, for every pound of muscle you gain, you burn roughly and extra 45-50 calories/day (as compared to that being a pound of fat)While it is hard to it muscle on, especially females, it cam be done. Ido believe you can simultaneously gain muscle and loose fat at the same time.
I do cardio and lift at the same time all the time. It isn't ideal but it says worked for me. I have read in more places than not, it is best to do cardio after you lift so that you have enough strength to really push the weights around. I prefer doing my cardio first as I feel it helps my muscle get really warmed up and I don't feel fatigued. However, with that said, I do try to switch it up frequently some body doesn't know what to expect...
I have a very intense workout schedule:
Monday's I take a spin class for an hour and do abs Tuesday I take boot camp and Pilates, Wednesday off, Thursday I currently lift chest/triceps/abs and do 20-30 minutes of cardio such as running, HIIT, elliptical trainer, Friday 60 minutes of spin plus back/biceps, Saturday legs/abs and 30-45 minutes cardio, Sunday shoulders/abs(if not done Saturday) and 20-60 minutes of cardio. I stretch everyday I work out for a good 20 minutes AFTER I workout. I feel it is a highly overlooked, yet imperative part, of a good fitness plan. I try to stretch every major muscle group.
Know that not one planets all. Everyone has different fitness goals. Right now mine is to loose body fat by about 3% and increase muscle. Good luck!