Note that you'll have to strip out any templates or infoboxes.
edit: If you want to extract the plain text (without wikilinks, etc), you can use the TextExtracts API. Use the available parameters there to adjust your output.
If you need to do this for a large number of articles, then instead of querying the website directly, consider downloading a Wikipedia database dump and then accessing it through an API such as JWPL.
There is actually a very nice prop called extracts that can be used with queries designed specifically for this purpose. Extracts allow you to get article extracts (truncated article text). There is a parameter called exintro that can be used to retrieve the text in the zeroth section (no additional assets like images or infoboxes). You can also retrieve extracts with finer granularity such as by a certain number of characters (exchars) or by a certain number of sentences(exsentences)
Please note that if you want the first paragraph specifically you still need to get the first tag. However in this API call there are no additional assets like images to parse. If you are satisfied with this introduction summary you can retrieve the text by running a function like PHP's strip_tag that remove the HTML tags.
You can download the Wikipedia database directly and parse all pages to XML with Wiki Parser, which is a standalone application. The first paragraph is a separate node in the resulting XML.
Alternatively, you can extract the first paragraph from its plain-text output.
The response you get is an array with the data, easy to parse:
[
"bee",
[
"Bee"
],
[
"Bees are flying insects closely related to wasps and ants, known for their role in pollination and, in the case of the best-known bee species, the European honey bee, for producing honey and beeswax."
],
[
"https://en.wikipedia.org/wiki/Bee"
]
]
To get just the first paragraph limit=1 is what you need.
You can use jQuery to do that. First create the URL with appropriate parameters. Check this link to understand what the parameters mean. Then use the $.ajax() method to retrieve the articles. Note that Wikipedia does not allow cross origin request. That's why we are using dataType : jsonp in the request.