Using a quote by someone or about something relevant to the topic of your paper is an effective and attention-grabbing way to start a paragraph. A proverb or well-known saying may also work just as well, but avoid being overly cliche. The best quotes are ones that will add to the argument you are making, so choose your quote carefully.
If you plan to discuss something that revolves around a key term, it may prove helpful to your reader to define it in the paragraph's opening sentence. Don't define words that everyone knows, but some words have multiple definitions and you may present one specific meaning to your reader.
A statistic, especially one that is surprising to the reader, will help to keep him interested in what your paper says and will drive him to continue reading. Like the previous techniques, it is important to keep the statistic relevant to the paper.
Use words to produce a mental image in the mind of your reader. If you are writing about a painting, play or something else visual, describe it. Capturing your reader's imagination and causing them to actively picture whatever it is that you are describing will hold their interest and make her excited to read on.
Analogies, metaphors and similes are effective ways to begin a paragraph, because it captivates the reader in a similar way as the previous step: by encouraging her to produce a mental image of whatever the topic of your paragraph may be. Keep the analogy, metaphor or simile relevant to the essay and avoid making it too over-the-top.