Whether you use "in" or "to" depends on how you are using word that comes after it.
When you use "interested in", you are usually talking about being interested in something (generally, a noun). For example, you can be interested in wine, or interested in cheese. Therefore, this sentence usually takes the form "[Someone] is interested in [something]."
There are times, however where the something you are interested in can be a verb. For example, you can be interested in swimming, or reading, or sleeping. In this case, the verb is acting like a noun. When a verb acts like a noun, it is called a gerund and almost always will end in "-ing".
"Interested to" is a bit more complicated, because by itself it doesn't really make sense. The "to" is actually part of the verb that comes after it, like in "to read", "to see", or "to hear". This "to form" of the verb is known as the infinitive, and is used to add detail to (or modify) the word that comes before it.
This type of sentence usually takes the form of "[Someone] is interested [to do something]."
To really answer your question...
"Interested in" is used when what comes after it is a noun, or a verb acting like a noun (known as a gerund).
"Interested to" is used when what comes after it is a verb in its "to form" (known as an infinitive).
"I am interested in starting my career in your company" is the preferred construction. While "I am interested to start my career in your company" may be technically correct, you should not use it because people generally don't use "interested" with "to start". A more commonly accepted way to say it while still using the "to" would be "I am excited to start my career in your company".
Hope this helps, and good luck with your new career! :)