[chryuop]

The fact is that it is not an easy question to answer. These things are different from case to case, so it it not easy to regroup them all like for English where you use "to" for everything.
Some can be done with the て form of the verb, some with the particle に, some use special words like ように、のため and others. Some again are expressed with auxiliaries which can be other verbs or adjectives.

In you second phrase for example the phrase "want to eat" is obtained joining the verb eat (食べる) with the adjective which express the desire (たい), so you will get that to say want to eat you will have to say 食べたい. If on the other hand you want to say "start to eat" you will use an auxiliary which in this case is a verb (I know 3 of them, I guess the most used would be 始める), thus you will have the phrase 食べ始める. Generally to show a purpose you will use the ます form followed by に and the main verb...you can have other complements in between (it is not like English where after the "to" you have to have immediately the other verb). So for example if you want to say "go to the shop to buy a new car" you can use this form 新しい車を買いに店へ行く (I have noticed that usually with this kind of form they use more へ for the location complement, but not sure if it is a rule).