What do you like best?
I love the integration with Github, when it adds visual information to a pull request to let you see how a build is progressing and what problems it has (if any).
I love that it's very easy to configure without any hassle.
I love the badge you can put on a website or in your readme to show whether the build is passing or not.
I love the fact that it has built-in support to deploy to other vendors, like Heroku
What do you dislike?
Difficult to build native .Net projects without using Mono, but that's more of a disappointment with the .Net build ecosystem and not a fault of Travis as such.
Also, pricing - I've only had need to use the free open-source tier, as all the projects I've worked with are open source, but the price of the first paid tier ($129 per month!) is, for me, quite prohibitive and really puts me off making the leap to pay for it.
If there was an even cheaper tier, for example one which was say $30 per month with 1 concurrent job and limited minutes per month, but allowed private repositories, then I would probably make that jump. I just feel there's a big gap in the entry-level pricing tiers.
Recommendations to others considering the product
Give it a try and not be put of by big words like "continuous deployment". It was so easy to use.
What business problems are you solving with the product? What benefits have you realized?
Developing a website for my wedding with another developer. We quickly realised the benefits of Continuous Deployment specifically, and more importantly, how easy it was to set it all up. Everyone should be doing this!