Another and other

Another means ‘an additional’ or ‘extra’. It is used with singular countable nouns.

Could I have another cup of coffee?

Other can be used with plural nouns.

Where are the other photos?

Arise and rise

Arise is commonly used with abstract nouns as subjects. It means ‘appear’, ‘begin’ or ‘come to one’s notice’.

Unexpectedly some difficulty arose.
A discussion arose about the best way to solve the problem.

‘Rise’ means ‘get higher’.

Prices keep rising.
When does the sun rise?

At first and first

At first is used to talk about the beginning of a situation.

At first everything seemed perfect.
The work was hard at first, but I managed to finish it somehow.

We use first in other cases.

I first met her at a library.
First we should discuss the history of the problem.

Note that at last is not the opposite of at first.

Bath and bathe

The verb bath is formal and is not normally used in American English. In an informal style, we say have a bath (British English) or take a bath (British and American English).

I am feeling hot. I should take a bath. (British and American English)
I should have a bath. (British English)

Bathe
In British English bathe means ‘swim for pleasure’. In American English, bathe is used to mean ‘take a bath’.

You should always bathe before you go to bed.

Note that bathe is rather formal. In an informal style, we say ‘have a swim’ or ‘go for a swim’.

Let’s go for a swim in the sea.