Bex,
first thing, go out and buy yourself a heart rate monitor. You really need to make sure you are controlling your effort properly to be able to build up your base fitness level. Garmin, Polar and Timex all do a whole range with some very reasonably priced basic ones, right through to several hundred pounds depending what you want.
If you are starting from scratch, throw any targets related to speed out of the window and just concentrate on fitness, the speed (if you want it) will come later.
A very handy formula for target heart range is the MAF formula, from a guy called Phil Maffetone, for Maximum Aerobic Funntion, very simply 180 - age, with a few adjustment factors depending on your personal circumstances. For you, as you are a complete beginner I'd go with 180-age-5. For me I go with +5 as I'm an experienced athlete, so that gives me 180-39+5 = 146.
When you go out jogging, do not let your HR go above this level, for at least the next 3 months I'd suggest. If that means you have to slow down, or even walk, then that's what you should do. The reason for this is that what you need to do is to develop a good base level of fitness. This will help in a number of ways; your metabolism will becoem more predictable which will make your response to insulin and food intake more predictable; it will help you understand better how you respond to exercise; it will improve your insulin sensitvity; it will allow you to exercise more regularly because you will not be overdoing it each time.
I suggest you pick a target time for exercise to start with, say 20-30 mins, and when your fitness is starting to improve, defined by being able to jog for all or most of that time, then start increasing duration or speed gradually.
I do all of my training based on HR, with the majority of it outside the racing season being controlled at these "easy" HR levels. Like you I'm new to diabetes, but not managing exercise. So far, I've not had to alter insulin levels for shorter sessions of an hour or less, but I have done for longer runs of 2-3 hours or more. What I've been doing is making sure I don't have any active bolus on board, so exercise before meals or 2-3 hours after. I always take my test kit with me, and test every 6km (around 30-35 mins at my steady pace) and take a medium acting carb source in case I find my level dropping. You need to keep testing to get a pattern of how exercise affects you.
One reason you may have seen the drop and then massive spike is becasue you are working too hard, hence the HR monitor advice. For me, I know that on a cycling time trial I don't need any carbs beforehand as the adrenaline response from the hard work will dump glucose from my liver and I can see a rise from start to finish, you may be seeing the same.
That's quite a lot to squeeze into one post and for you to understand, feel free to ask any more questions or even drop me a PM.
Dave