Inflating the wings. As soon as it emerges, the butterfly pumps fluid from its abdomen into channels in its wings, literally inflating them. As the wings enlarge, the abdomen shrinks.
Shifting its grip. As it emerges, the butterfly first grips with its claws the ridge around the top of the shell, a structure that probably evolved for this purpose.
Shifting to something like a twig provides a better grip. It's also possible shifting evens out the forces on the leg as they harden, assuring they assume the right shape.
Feeling with feet. When they are getting ready to fly or want to fly or move to a better location, butterflies reach out with their feet to find new footing.
Twisting. The butterfly often rotates its position while hanging from the chrysalid shell or nearby twig. I don't know the reason--it may be trying to maximize heat exposure to the sun so it can warm, preparatory to flying and to speed hardening. Slight changes in position and in the direction of gravity might assure the wings become perfectly shaped.
Startle response: Giving a rapid jerk or wing flap when you approach is a sign of alarm. The butterflies are programmed to avoid unfamiliar large objects. But they rapidly get used to people (habituate) if no harm results.
Excretion of an orange liquid called "meconium." This happens roughly half an hour or hour after eclosure, several times. This is the waste (analogous to urine) left over from metabolism in the chrysalid. Protect the surface under the butterfly with a paper towel.
Warming flight muscles before flying. They may slowly flap their wings in a full stroke, or move them rapidly with a short stroke (like shivering in humans). These behaviors signal impending flight, perhaps within ten minutes or so of starting the warmup.
Turning the head both ways. This is a sign they are ready to fly. Most of the time, they will actually fly within a second or so after looking both ways. Just enough time to put them back into the mesh cube!
Extending the proboscis. Immediately after starting eclosure, the butterfly "zips together" the two halves of its proboscis. By extending the proboscis, it may be testing that the proboscis is working properly. A butterfly also extends its proboscis when it tastes something sweet with its front foot.
Feeding. The butterfly pumps sugar solution or nectar through its proboscis, often for many minutes at a time. Newly emerged butterflies have instinctive preferences for certain colors and odors, but they are able to quickly learn new preferences, depending on what nectar sources are blooming.
Flight. The first flights are usually a little weak, but strength and flight ability build rapidly. By the afternoon (in proper conditions), the butterfly could probably fly a considerable distance.
Direction of flight. Butterflies indoors always fly towards light. If you release them outdoors in the afternoon, they will likely fly towards the sun, to the southwest. Usually, they fly upwards and land in a nearby tree or bush. Wild butterflies rest, bask, ride out bad weather, or spend the night in trees. So, landing in a tree located in the same direction as the sun is expected--not an indication it has begun migration towards Mexico.
If you had just been "born," and suddenly tossed into the air by a huge monster, you'd probably want to rest in a safe place for a while, snack, and get your bearings, before starting on a journey of 3,000 miles.