What are some marketing strategies that you have used in the past or that have been proven to work well to obtain your competitor's customers and gain more market share?
First it depends on the platform that you are marketing on. Mobile or desktop? There are a lot of way as mentioned below my answer. You can buy ppc ads against your competitors keywords on google. You can run targeted ads on Facebook that target people who are connected to your competitors. There are actually some grey area platforms that can data mine social platforms as well and you can use them to create Look A Like/Custom Audience clusters which you can use to target. These are just a couple ways to hit your competitors customers.
At the end of the day if you are looking for paid specific acquisition/marketing it is expensive when you are competing against the larger companies because typically they are buying traffic on a much more aggressive eCPM for relevant traffic. Again understand your competitor but I recommend as a small to mid sized business focusing on organic traffic and building more competitive social components in your strategy so that you have a strong viral k-factor. Ensure that your existing customers are strong and loyal so that they do all the advertising for you. (E.g referral traffic). You will end up eating their market share incrementally provided your business is retaining customers.