My usual first night is to obtain wood as almost all people do. My first night shelter is usually built near water by pillaring up 5 or 6 blocks and building a platform with ladder access up the pillar. You can also have a ready made version of this by finding a good sized tree and clearing a one block column of the branches vertically adajcent to one side of the trunk and putting ladders up the trunk and sitting up top on the tree. I like these bases because you get a good view of the surrounds and at day break can jump down into a safe area and hopefully obtain some bones from disintergrated skeletons - as i usually plant wheat early on near the water, the bone meal can speed up the wheat and get bread quicker. Since the addition of Phantoms, probably important to make a roof on these kinds of bases now. Also, the doube as beacons when lit so you can find your base from distance.
If you're gonna kill animals for food, make sure you leave at least two for future breeding. Thats vital i think.
I usually mine on the 3rd or 3th day, but more usually i get more wood and replant saplings near and around my base and plant more wheat first two days.
My first main base in my current world still has my original first night (and a few good nights after whilst i worked on the base) elevated platform as below. The first pic was a few nights in to the world, the second pic was 10000+ days later
I've also built first night homes in the side of a hill or something, preferably close to or in water for planting wheat. I can mine into the hill or down at night.