How do you naturally say these sentences in Persian?
To say "I want to eat", I know you use the subjunctive form and it looks something like this:
می‌خواهم بخورم
But how do you form these other sentences with infinitives in them?:

I need to eat
I started to eat
I learnt to dance
I started to learn to dance

These are my guesses? Any correct?
خوردن را لازم دارم
خوردن را شروع کردم
رقصیدن را یاد گرفتم
رقصیدن را یاد گرفتن را شروع کردم

Also, are any of these alternative sentences natural?:
شروع به خوردن کردم
شروع کردم به خوردن
شروعِ خوردن کردم (with ezafe)

Any hints in general to making some of these sentences? Or what is the most casual and most formal ways of saying these things?

Thanks!

Question

How do you naturally say these sentences in Persian?
To say "I want to eat", I know you use the subjunctive form and it looks something like this:
می‌خواهم بخورم
But how do you form these other sentences with infinitives in them?:

I need to eat
I started to eat
I learnt to dance
I started to learn to dance

These are my guesses? Any correct?
خوردن را لازم دارم
خوردن را شروع کردم
رقصیدن را یاد گرفتم
رقصیدن را یاد گرفتن را شروع کردم

Also, are any of these alternative sentences natural?:
شروع به خوردن کردم
شروع کردم به خوردن
شروعِ خوردن کردم (with ezafe)

Any hints in general to making some of these sentences? Or what is the most casual and most formal ways of saying these things?

Thanks!

Answer

Hello Tom!
this is one of the common questions which Persian learners usually ask.
As you know how to say I want to eat, می خواهم بخورم, you can see there is a ب added to the verb. these kind of verbs are called مضارع التزامی. they're used after the some verb like: want, need, love, like, know and etc. generally in some cases which two verbs are connected together. 
Ex: I need to eat = من نیاز دارم که بخورم. of course this sentence is not common. "needing" shows wanting something very much so it's better to say میخواهم بخورم. 
Ex2: I started to eat = شروع کردم "به" خوردن . the sentence you mentioned خوردن را شروع کردم is also correct but not very common. 
Ex3: I learnt to dance = رقصیدن را یاد گرفتم, is correct. and if you want to say I learnt how to dance you you should say " یاد گرفتم چطوری "برقصم. 
Ex4: I started to learn to dance: شروع کردم که رقصیدن رو یاد بگیرم or شروع به یادگیری رقص کردم
شروع به خوردن کردم
شروع کردم به خوردن
theses sentences are totally correct. but the third one is wrong.

Answer

Let me make you another example: 
I love foreign languages, so I need to learn different languages. recently I started to learn German.
من عاشق زبان های خارجی هستم. بنابراین می خواهم زبان های مختلفی را یاد "بگیرم". به تازگی شروع کردم به آموختن زبان آلمانی. ( یا شروع کردم که زبان آلمانی را یاد بگیرم)

Answer

0) The verb "to eat" can be used either with or without an object in English. Not the case in Persian. You always need an object with "خوردن".

1) If talking grammar only:
I need to eat ≈ (niyâz dâram [ke] bekhoram) نیاز دارم [که] چیزی بخورم
I started to eat ≈ (shoru' kardam [ke] bekhoram) شروع کردم [که] بخورم
I learnt to dance ≈ (yâd gereftam [ke] beraghsam) یاد گرفتم [که] برقصم
I started to learn to dance ≈ (shoru' kardam [ke] yâd begiram [ke] beraghsam) شروع کردم [که] یاد بگیرم [که] برقصم

2) Most of these are expressed slightly differently by native speakers though, such as but not limited to the following:
(bâyad chizi bekhoram) باید چیزی بخورم ≈ I must eat
(shoru' kardam be khordan) شروع کردم به خوردن ≈ I started eating
(raghsidan râ yâd gereftam) رقصیدن را یاد گرفتم ≈ I learnt dancing
(shoru' kardam raghsidan yâd gereftan) شروع کردم رقصیدن یاد گرفتن ≈ I started learning dancing

Answer

Ezafe is not only using "e".
That u thing always u have to use it,and remember in persian we use the verb at the end of the sentences...for instance u can say;

.شروع به آموختنِ (یادگیری) زبانِ المانی کردم

집에서 편안하게 언어를 배울 수 있는 기회를 놓치지 마세요. 경험 많은 강사진을 살펴보고 지금 바로 첫 수업을 신청하세요!