When I started on rollers I did my first rides in trainers as it meant I could put my foot down should the worst happen. A handy little confidence booster. I'd also recommend you use a kitchen worktop, high backed chair, or similar, rather than a doorway. This gives you a single safe side, not that it matters since you're wearing trainers, and it gives something to put your hand down on, which you can't do with a door jamb. I think this is more natural than gripping a vertical surface, and allows you to hover your hand above it if needed.
Keep your hands on the tops. You can use the drops, hoods, or tribars, but using the tops of the bars is easiest.
Don't have a white knuckle death grip. A light touch is all that's required. Thanks to the wonders of physics, it's not actually necessary to hold the bars at all provided the wheels are rotating at a reasonable speed.
Look forward. Relax. Smile. Pedal. As illustrated by the rider below.
"Here's how you run a marathon. Step 1: You start running. Step 2: There is no step 2." - Barney (How I Met Your Mother)