I can't really give you an overview of the market, but I can list a couple of devices that I've got, or my wife has, to give you an idea of what is around. But I'll say straight off that this might be limited value to you.
First there is the Garmin Edge. This is ideal for measuring cycling data - you have external sensors such as heart rate monitor, speed, cadence, power sensors (which connect via ANT+). The 800 series and above are top of the pile, with maps'n'all (and support all the sensors). The 500 series support routes but don't display maps - they also support the sensors. Garmin have introduced some lower-end devices but double check before you go and buy - some of them don't support the external sensors so for your purposes would be....limiting.
I'd probably say that this range of devices is pretty useless for activities other than cycling.
Also in the Garmin range is the Vivofit. I was given one of these so also have experience of it. It is basically a step counter. I'd imagine it would count steps while jogging, but bear in mind that it is just a counter. The Vivofit will also measure your heart rate, but it does so by an ANT+ connection to the Garmin Chest Strap. You'd think they could have come up with something better.
My wife has a Fitbit. As fas as I can tell from having talked to her, it is the functional equivalent of the Vivofit. There are nuances with the displays (all are designed to be minimalist so the display on all is non-existent or limited). Apparently the new Fitbit will also measure your heart rate (the older ones don't), simply by virtue of being on your wrist. I'm pretty sure you can forget about using any Fitbit devices for cycling.
Mt wife and I did also look at an exercise computer for her, and we came across this Forerunner 920XT watch that @GlennGervais mentions. Double-check this, but the beauty of this device is that it supports all these ANT+ sensors, so can capture anything that the bike computers can capture. I also know that there is another ANT+ sensor designed to fix onto your trainer (shoe) so you can get metrics when jogging. Presumably things like step distance, speed and whatever the running equivalent is of cadence.
The downside with this device is the price - it is currently on Garmin's site at over GBP400, so you really are talking top of the range here. (And don't forget the sensors will be extra.) Also, I'm not sure how it attaches to the bike - with the Edge computers they sit on the handlebars and are pretty much ideal.
In terms of other devices, I know the Samsung S5 and the new iPhone have both gone down a personal fitness route, so there may be some mileage in that. Of course, these phones themselves cost a pretty penny to start with. But certainly some of the cycling web sites (e.g. Garmin Connect, Ride With GPS, Strava) offer apps to allow you to capture ride data, so if you have a fancy phone in any case these could be realistic options for you.
That's about the limit of my knowledge. Hopefully other people can chip in with information about other options - it wouldn't surprise me if people like Polar had the exact same devices on the market as Garmin, although I'd also expect the prices to be similar.
When you look at the notion of a single fitness device that measures everything, this is very early days.