てごらん

Gently urges the listener to try doing something, much like "try ~ing" or "have a go at ~". It is typically said by adults to children or by seniors to juniors.

Spoken and warm in tone. It is used by parents, teachers, and other seniors to children or people of lower status, often to encourage them or to draw attention to something worth experiencing. Unlike the plain command なさい, it sounds kind and encouraging rather than strict. Unlike たらどう, it invites the listener to try something rather than offering advice.

• てみるN4「てみる」is stylistically neutral and works for yourself or others; it inflects into forms like 「てみたい」, 「てみてください」, and 「てみた」 to express a wish, make a request, or state the result of having tried something. 「てごらん」, by contrast, is used only when a superior urges a junior to give something a try.
• たらどうN4「たらどう」attaches to the ta-form and suggests in question form that the listener do something, not limited to trying; adding 「ですか」makes it more polite, and a heavier tone adds a reproachful edge. When urging someone to try, 「てみたらどう」is close to 「てごらん」, but the latter is restricted to superior-to-inferior use.
• なさいN4「なさい」attaches to the ます-stem (the ます form with ます removed) and expresses a superior's command or urging, and it also appears in exam instructions; 「てごらん」attaches to the te-form and warmly encourages the listener to give something a try. Neither can be used toward a superior, and switching to 「なさい」turns the invitation into a command.
• ばいいN3「ばいい」attaches to the ba-form and points out the way to a goal—'just do this and it's fine'—in a neutral register with no hierarchy restrictions; it can also express a wish, regret, or asking for a solution. 「てごらん」attaches to the te-form and is a superior-to-inferior prompt to 'give it a try,' so the two should not be swapped casually.
• てみせるN2「てみせる」means doing something so others can see it (a demonstration) or declaring the speaker's resolve to definitely get it done, in a neutral register; 「てごらん」is a colloquial, superior-to-inferior prompt for the listener to try it themselves. Both attach to the verb's te-form, but their meanings differ and they cannot be interchanged.

てみる is the neutral "try doing" and can describe anyone's attempt, whereas てごらん is a gentle top-down invitation aimed only at the listener; なさい is a firmer command, and たらどう and ばいい give advice or suggest a solution rather than inviting someone to experience something. てみせる means doing something to demonstrate it or to prove that one can, which is a different idea entirely.