begin to do一般译为"开始做```',用于表示主语之前没做这件事 
eg;After finishing the homework,he began to watch TV.
begin doing 一般译为”开始（继续）做”,用于表示主语之前就在做的某件事,中间有小小停顿,现在继续开始做．
eg;He was doing homework when the telephone rang ,it was mother,after answering it,he began working again
begin doing 多注重于直接开始做某件事,而begin to do 是多注重于重一种状态过度到另一种状态 
一般来说,begin to do和begin doing可以互换,但在以下三种情况下,用to do：
1.主语不是指人,而是it等.如：It began to rain.
2.begin后接表示心理活动的词.如：begin to know 
believe,wonder,think等.
3.begin本身是ing形式,为避免重复后接to do 
即,beginning to do 
P.S.
doing 一般有正在做的意思 
to do一般有正准备做的意思 
所以doing 就是开始做了to do开始要做,还没做
另外和begin类似用法的还有start:
start doing 和start to do的区别 
1)当我们谈论一项长期的或习惯性的活动时，用doing形式的情形较多。 
How old were you when you first started playing the piano? 你最初弹钢琴的时候有多大？ 
*比较She sat down at the piano and started to play. 她在钢琴前坐下开始弹了起来。 
2)start 的进行时态之后，不用doing形式。 
①I was starting to get angry. 我开始发火了。 
②I am starting to cook the dinner. 我正要开始煮饭。 
3)在 start 之后出现的动词涉及感情或思想(如 understand, realize等)时，只可用不定式。 
①She started to understand what he really wanted. 她开始明白他真正想要什么。 
②She started to wonder why he was here. 她开始觉得奇怪为什么他会在这儿。 
4)当主语是物不是人时，要用不定式。