Gym bunnies with type 1.

An old gym bag here! You need your insulin, test kit and hypo stuff just in case. Some people carb up before hand. Some people reduce insulin before if they are eating.
The activity can either raise your sugars or lower. In my case weights and HIIT raises sugar so I increase the basal on my pump by +50% half an hour before. I quite often need to bolus as well roughly 1 unit.
It's different for everyone. This type of exercise can cause fast snatch back of sugar as well afterwards. I've been caught out on that.
Cardio lowers my sugars so I reduce my basal by -50% half hour beforehand. Again the affects can be a couple of hours afterwards.
The best thing you can do us to test regularly. Cos I'm such a scaredy cat I test every 15 minutes during cardio.
I aim to be in the 5s for weights and in the 6s for cardio. Testing and correcting as needed.
Hope it helps.

Going to the gym will not make you healthy by default though generally considered any form of exercise is better than none.

About the insulin spiking: Insulin is basically an anabolic hormone. Meaning that insulin will aid in the development of muscle. Your muscle cells have insulin receptors and when insulin reaches these cells it opens them up to "receive" nutrients (amino acids, creatine, ...) that are necessary in the development/rebuilding of muscle. Furthermore it is required to start protein synthesis. This is the reason why bodybuilders tend to spike insulin. In short it increases the rate at which muscle absorb the nutrients required to grow. Insulin is also on the list of illegal drugs for competition sports, rules probably differ from country to country but i always had to have a prescription for both types of insulin and a signed official form from my endocrinologist in case i got a blood test. Since your pancreas isn't functioning as it should you do not have to be worried with insulin spiking since you have access to insulin. So if you take a protein shake for example after a workout and you know your BG rises from it you can add some insulin. If not you already got enough insulin in you.

Also you should never leave the house without insulin so yes take it with you and you should always be able to monitor your BG. If you're uncertain of the effects of your exercise regime on your BG then you can measure it before and after for a couple of days/weeks to see the effects. You can also have a rise or drop a significant time after exercising dependant on the type of workout. For a lot of people, myself included, trial and error is an annoying but necessary process.

Lastly getting fit is not as important as staying that way so don't rush it and take your time. Eventually you will succeed in your goals if you keep at it.