You may be mixing up two different ideas: Western vs. non-Western and developing versus developed nations.
'Western' refers to a broad set of cultural - and religious - features that have historically served to divide Greek, roman and - later - judeo-christian societies from their neighors and enemies. In more recent times, 'Western' refers to a judeo-christian religious past and adherence to certain ideas about how societies should be structured and governed - such as the idea that democratic governments with universal sufferage are preferrable to absolute monarchies. The term 'Western' is not always clear and is very much a rhetorical invention rather than a coherent cultural or political category.
Nontheless, Mexico would clearly be a part of the 'West.' It is a primarily Christian nation that officially speaks a Western language and has political and social institutions that are historically linked to other nations in the 'West.' It has a non-Western past that continues to exercise influence over contemporary society; still, as a nation, Mexico is clearly Western.
The other idea is 'developed' versus 'developing' nation. Like 'Western,' 'developed' refers to a broad set of ideas about the role of capital, industry and economic institutions in a given nation. 'Developed' countries are those with complex industry (instead of just agriculture), capital markets (based on currencies instead of barter) and labor (which is skilled and free to accept or reject employment) that are not only advanced in a historical sense, but are also reasonably independent of the state. 'Developing' nations are those where industry may be more geared toward agriculture or raw materials, where capital markets may be unstable (for example, because banks are not independent of the government, or the government prints more money when it runs low) and where labor is either largely unskilled or has less complex skills than would be the case in industrialized "developed" countries.
In this way, Mexico is both a Western and a developing country.
There are countries that are non-Western and developed (e.g. Japan), as well as non-Western Developing countries (e.g. Indonesia).
Hope this helps.
By the way - the whole 1st, 2nd and 3rd world thing is even more confusing. These are cold war terms that refer to:
1st world: market oriented countries (e.g. the US, Canada, Western Europe),
2nd world: socialist countries with state controlled economies (e.g. the old Soviet Union, China), and
3rd world: non-aligned countries that might have allied themselves with either "side" of the cold war (e.g. India, Brazil, Mexico, Indonesia)