Start to do / Start doing

The difference concerns more the focus on the action after the word start than on the start itself.
The same rule is used for begin and finish.

Использование инфинитива или герундия после слова start зависит от того, на чем нужно сделать акцент - на действии как таковом, или же на начале самого действия.

Это же правило используется для слов begin и finish.

Start + Infinitive	Start + Gerund
Start to do focuses more on a single activity, starting point of an activity, denotes more spontaneous actions, unprepared actions, indicating the very beginning of the action rather than its process or continuation. Start to with stative verbs is more preferable: start to feel, start to seem, start to appear, start to sound, etc. Сосредотачивается больше на самом действии, отправной точке деятельности, обозначает более спонтанные, неподготовленные действия, указывающие на самое начало действия, а не на процесс или продолжение. Start предпочтительнее использовать со статичными глаголами: start to feel, start to seem, start to appear, start to sound, etc.	Start doing focuses more on continuing activity in the process, denotes prepared actions, not the actual start of the action. Больше фокусируется на длительности процесса, обозначает подготовленные действия, а не фактическое начало действия.
I'll probably leave them up until the weather starts to get cold. If you start to care at last, you are welcome home. Why do not you start to drive, I am tired of waiting. I am starting to get bored. Will you ever start to breakfast in the mornings? I started to follow him up the stairs… I started to feel neglected. Looking at her voluptuous body, men start to get ideas. Our work starts to make sense. He stopped walking when he saw her start to shudder.	Do not start acting like a child. He was so angry, that started throwing things at her. Do not pretend you will start crying. Start learning what it means to be in charge of something. Zip your mouth. You have started spouting some secret information. If you start yelling again, I will throw something at you. Teacher asks children to start reading. I do not care to start making the speech first. After all these events we started living in fear. She started cleaning the kitchen with tears in her eyes.