I suggest you to try out the Mi Band 2 by Xiaomi. It's very cheap (I bought it for less than 20€ on Gearbest, even if it took 3 weeks to arrive from China to Italy). It has a small screen that shows the time, your heart rate, total steps and burned calories. It can also track your sleep, giving you a feedback. It's waterproof so you can wear it while you take a shower (however, don't use it while diving or swimming). The battery lasts for more than 20 days with a single charge.
Using its companion app, Mi Fit, available for Android smartphones and iPhone, it can show you notifications for incoming calls, messages as well as all the other apps you want.
It's very small, so you will almost forget you are wearing it.

I know there are lots of different fitness bands, but they are very expensive and offer the same functionalities, so I suggest you to buy a cheap one...