PT here
Do some shoulder and chest mobility first, to warm up the synovial fluid at your joints. So: stand up and cross your arms across your chest, sort of hugging yourself, then swing arms out to the sides, keeping your arms roughly shoulder height or just below. As you bring arms back in, alternate whichever arm is on top as you cross your arms across your chest. (I feel like I'm not explaining that one very well... it's one I usually demo rather than write down!)
Assuming no shoulder injury/impingement, elbow issues or similar, with a press-up it's better in the long run to do as much of the full range of the movement as possible. So it that means you do them on the knees, do them on the knees. I'd generally prefer to see a client do 3x12 knee press-ups than half a dozen little dips.
As pp say, doing them on a plyo box could help too. You'd put your hands on the box and feet on the floor. Don't do it the other way round, that makes it much harder!
You'll find that as you get stronger, over subsequent weeks, you can gradually position your knees a bit further back, which incrementally works you towards a position where you can do them with straight legs/knees raised.
You could also try doing chest press or bench press with dumbbells, a barbell or resistance bands to train the chest muscles, and shoulder presses to train shoulders, which will help with press-up form and strength.
Remember to focus on maintaining a strong posture through your core and middle; don't let your stomach drop down, because it puts unhelpful pressure on your lower back and pulls your shoulders out of alignment. And keep elbows tucked in to your sides as Klara says.