When a card transforms, its printed abilities and characteristics change to match its new form, but it is otherwise the same game object. That means the new form does not "enter the realm" and does not trigger any "enter the realm" effects like Genesis - it was there all along! Furthermore, since the new form doesn't enter the realm, it will not suffer from summoning sickness, assuming the old form didn't enter the realm this turn.
If one card transforms into a different card, the original card is banished but game effects treat it as the same game object. For example, if you play on my , I end up with a Frog token under my control and the Apprentice Wizard card is banished. If some game effect was tracking the Apprentice Wizard, it is now tracking the Frog token.
If one card transforms into a different type, e.g. , continue to use the same card.
When a card transforms, it does not die and deathrites do not trigger. For example, if you my into a Frog token, I do not resolve the Leprechaun's deathrite.
When a card transform, control of the card does not change unless specified by card text. For example, if I use on your site, you still will control the new . If you summon to transform one of my artifacts, you will still control your Mimic because of Mimic's specific text, "under your control."
As mentioned above, the printed abilities and characteristics change to match the new form upon transformation. For example, if a is ed into , it no longer has Airborne or 3 power, and its Genesis does not trigger. Instead, it has Submerge and 16 power.
In contrast, any ongoing effects continue to apply to the new form! For example, if I cast on my and then I it into , it does have Airborne until my next turn, since Power of Flight applies to the game object and isn't a printed ability.
The game treats the new form as the same game object. For example, if the old form was tapped, had taken damage, and was carrying an artifact, the new form will also be tapped, have damage, and will be carrying that artifact.
Note that Stealth and Ward, though a printed keyword on many cards, are tracked via a Stealth mark and a Ward mark, respectively. Therefore, Stealth and Ward marks will carry over to the new form. For example, if I a with a Stealth mark into a , it will still have the Stealth mark. If I cast on my and then it into , then it will also still have the Stealth mark. Unlike the Airborne example, Stealth and Ward are tracked via a mark, which carries over.
Finally, if you Shapeshift a minion into a new form with the printed Stealth or Ward keyword, Stealth and Ward will not trigger since they are an "enters the realm" effect. You only get the Stealth mark or Ward mark when the minion enters the realm, and it's not entering the realm when transforming.