What is the proper way of using "other"? [closed]

It depends on context (as is so often the case). You can say "Two other methods are..." and you can say "The other two methods are..." however using "Other two" without the definite article is not usually correct.

It also needs pointing out that "Two other methods are..." implies that the methods to be described are only two of many whereas "The other two methods are..." implies that the methods to be described are the only alternatives to the one(s) already described.

For example if you describe method A as the most common or most efficient then say "Two other methods are B and C" you are implying the possible exisitence of method D as a minimum and the potential existence of many others such as E, F, G and so on.

If, on the other hand, you describe methods A and B as the most common ones then say "The other two methods are C and D." you are saying that E, F, G and so on do not exist.

"Other" is an adjective and so can't be placed before a determiner or number (two); "other two methods" needs a determiner (the, any, its, which, …);

numbers: Like all determiners, numbers come at the beginning of a noun phrase, so they come in front of any adjective(s).