One of the most important things is camera stabilisation.
A good, fluid head tripod is a necessity for any form of semi-serious video. For less 'locked down' requirements there are a range of solutions from monopods (my favorite for weddings) to shoulder mounts & steadycam rigs. All have their good and bad points so if you can borrow any of this gear from someone else to try out it might really help you figure out your shooting style and gear preferences.
Once you have figured out how to keep the camera stable, you can then start to think more about utilising motion effectively - not just superficially. Things like motivated pans (eg panning along a landscape in the direction traffic is flowing) & reveals (panning/moving the camera to reveal more information in a shot, such as another person or object) go a long way towards making the content more engaging, rather than just waving the camera round at anything that moves.
Also try to think about sequencing - eg starting with a wide shot, to show location, then a mid shot to shot the character, then CU to show what they are doing/feeling etc. You'll also need reaction shots (if you're shooting a footaball game and somebody gets a touchdown, show us the crowds reaction) B-roll (overlay which shows vision of what someone is talking about) and cut-aways (shots of detail in a scene can be useful to reveal more info and to cover awkward edits later on - for example if you accidentally "crossed the line" while shooting. The important thing to remember with sequencing is that you don't have to shoot it in order - but think about it the whole time and keep a mental not of what you do and don't have.