I don't know that you will be able to find anything that will automatically know if you're walking, running, or riding. The easiest would be a high-end wearable like an Apple Watch or Garmin Fenix, where it's always tracking you, and you can change modes with a couple of taps.

I have a Garmin VivoFit, which is like their version of a Fitbit. It does a pretty good job of discerning walking from running, and since it's inertial, rather than GPS, it works indoors and out. Syncs with the phone to get miles/calories in to the Health Apps.
I don't believe that any of the wearables will do cycling on their own, unless they have an integrated GPS, or are synced with a phone.

I use an app ( currently RideWithGPS) to record mileage on my rides, but I mount a basic computer on the bike as a speedometer, even a $10 one works fine, if you take the time to set it up right.
It's on as long as the wheels are turning, and the batteries last forever. On long rides, of several hours, it saves tons of battery life on the phone, since you don't need the screen on at all times.

As far as accuracy, when you're using phone apps, it all depends on your phone. My old HTC wasn't bad, but the iPhone SE that replaced it had a horrible time with GPS dropouts, so much that I stopped tracking with the phone, and just went with the wristband. The 5 got broke, so I got an old iPhone 4S out of the drawer, and found it had great signal. RWGPS can even tell which side of the street i'm on. YMMV