I don't think there are any hard and fast rules as to what words you should and shouldn't use to start a paragraph. It also depends what kind of document you are writing. In fiction and poetry or free verse you can be a lot freer with your use of language, and write in a style which is your own, even if it does not fit within tradition constructs of how language should be used. But if you are writing an academic essay for school or university, you will be expected to write in a way which is a great deal more formal.
Although a new paragraph is often a way of starting to discuss a new topic, in order to maintain the flow of the piece as a whole, you may want to look at using "linking" words or phrases sometimes, so that each paragraph does not sit on the page independent of those that become before and afterwords.
Some examples of these type of phrases are:
A contrary explanation is that, …
Although, …
As a consequence, …
As a result, …
However, I wouldn't use these each time as it could be quite repetitive and make your writing seem clunky.