Why not another ideas?

Another is always singular: it means an additional one. For more than one, use other: other ideas.

When do I use the other instead of another?

Use the other for something specific, already mentioned, especially the second of exactly two: one was late, the other was on time.

Grammar, compared

Another is one more, singular. Other is for things already mentioned or for plurals.

Another / Other
Another	Other
Can I have another coffee? (one more, singular)	I don't like the other coffee, this one is better. (a specific one already mentioned)
Try another approach. (one more, any)	The other approach failed. (the specific one already known)
She has two dogs. One is calm, the other is wild. (the second of two)	There are other ways to solve this. (plural, general)

Another and other compared
Point	Another	Other
Meaning	One more, an additional one	Different, or already mentioned
Form	another + singular noun	other + plural noun, or the other + singular
Example	Can I try another idea?	Do you have other ideas?
With exactly two	Not used this way	the other (the second of two)

Otro covers both in Spanish, so English asks you to check: is it one more, from an open set (another), or a specific one you already know about (the other)?
With exactly two things, use the other for the second one: one twin is tall, the other is short.
Another is always singular: another idea, not another ideas.
Other needs a plural noun or the before a singular: other people, the other day.

Why not another ideas?: Another is always singular: it means an additional one. For more than one, use other: other ideas.
When do I use the other instead of another?: Use the other for something specific, already mentioned, especially the second of exactly two: one was late, the other was on time.

Book a trial class. She writes back with times in Buenos Aires time.