I've used some chargers that were okay when the phone was off, but if the screen was on, in use for anything, they couldn't manage.
Chargers used to put out only 500 mA (the limit of USB 2 ports, IIRC), but most phone chargers should put out 1000 mA (1A). Tablet chargers are capable of even more. The phone's chargers circuits will take what they can, so you won't damage plugging your phone into a good quality tablet charger. Look for the fine print on the charger by "OUTPUT."
If you use a USB port on a computer, try to use a USB 3 port, preferably one with a power symbol by it, indicating it is meant to provide charging (some even do it with the laptop in sleep mode).