I have a Sigma Sport and can use it for running and cycling. And as others have mentioned, your HR will be a bit higher for running. But that doesn't mean you aren't getting a good workout from cycling.
So basically there are two parts to a HRM. There is a chest strap and a receiver that looks like a large dorky watch. The chest strap shouldn't move around at all if you have it adjusted right. It is surprisingly comfortable. The watch part can be put on your bike with the included mounting device or can be worn as a watch. The Sigma sport can be worn in the pool, but NOT after you change the battery. It WILL leak. You have to take it to a watch shop or jeweler to get it changed.
And if you aren't getting a hard workout on the bike you aren't going fast enough. Do some hills, go longer, or do some intervals...or do all three!
What do you currently ride? Time, distance, average speed, cadence, terrain, wind conditions, etc...?