There are a number of ways. This is the right option because you are talking about more than one thing.

To add to the previous answer: according to Hugh Dellar, some people tend to say 'there's' even with plural nouns especially when they use 'a lot': there's a lot of rain, there's a lot of policemen just because it's quicker ans easier than 'there are a lot of'. However, you'll be penalised for that in exams.

In informal speech, many native speakers would say 'There's a number of ways'. This is easier to say than the plural form, and it sounds natural. I use this form myself in relaxed speech. It also makes some sense grammatically, too, because of the singular noun 'number'. It's like saying 'There's a lot of...'. It really doesn't sound too wrong to the native speaker's ear. That said, the more grammatically correct form is 'There are a number of ways', and if you are doing a grammar exercise or writing a formal text, you should use the plural verb form.