Recent versions of Shadowrun have not gone into any significant detail on what Shamanism looks like, because it is explicitly based on RL Shamanism. The idea is that practicing Shamans suddenly found their rituals and rites were producing obvious magical effects after the Awakening. And then many were approached by the actual Spirit Totem they had already been following prior to the Awakening. The only thing that really changed is that their spirit totems taught them new rituals to produce new effects (new spells), but all followed the same patterns they were already using.
(In fairness, Shadowrun doesn't go very deep into what most practices look like for this exact reason. The Shinto tradition mimics RL Shintoism, same with Vodou, Wicca, Hinduism, Christian Theurgy, and basically all the rest)
Thus, Shamanism in Shadowrun looks a lot like Shamanism does in the real world. Because that's exactly what it is...Real World Shamanism, that has obvious magical output.
So, to give a very brief primer on what that might look like...
Shamanistic rituals encompass an extremely broad variety of actions and behaviors, partly because 'Shaman' is a term applied everywhere from Mongolia to the Native American tribes. Half a world away, there are obviously going to be enormous differences in how they work. But there are a few common threads...
Shaman often follow ritualistic behaviors, even when not trying to do magic. This may include restrictions on diet, behaviors, and so on. Some of this is purely practical, because some of the psychoactive substances used in shamanistic rituals can react very badly with certain kinds of food. For example, Tyramine, a compound found in many fermented foods and preserved meats, reacts very poorly with the active compounds in an Ayahuasca brew, and can cause a massive, potentially lethal blood pressure spike.
As for actual 'spellcasting...'
Contacting spirits would generally be done by way of meditation, often surrounded by incense or, at times, psychoactive substances. In essence, to 'contact the spirit realm,' the Shaman puts themselves into a trance either via a form of autohypnosis, or using psychoactive compounds to induce the state.
Casting 'spells' could be interpreted through the lens of the more active shamanistic rituals which may include dance, music, chanting, and other such things. If the particular shamanistic tradition in question includes Totem Animals, then the movements during these rituals may be intended as a mimicry of that totem. Steady stalking motions for a Wolf, broad movements of the arms for an Eagle, and so on.
So, I clearly can't provide a full primer on RL Shamanism here, it's far too broad of a subject, but hopefully this is enough to get you started. Beyond this, I'd recommend hitting up Wikipedia and Youtube.
But, just to give an example...
Suppose two spellcasters were trying to cast a Fireball spell...a hermetic caster and a shaman. (Note: This is me-generated flavor text....descriptive, not prescriptive)
The Hermetic Mage would perform very precise movements, recite a precise magical formula, perhaps offer specific material components, and use these to shape his magic to create the fireball. Because long research and study has shown that this is how you cast Fireball.
The Shaman would perform movements intended to evoke the idea of Fire, possibly chanting and vocalizing in a way that sounds reminiscent of the crackle of fire, intended to call the Spirits of Fire to come together into a weapon that he may cast at his enemies. If he uses reagents, it would be something like leaves or dried wood...something that fire would catch on quickly. The spell might not even look exactly the same every time it is cast...