Good morning, Wonderdome!
Great question. Very briefly:
Increasing internal links typically depends on having a vision for how the content on your website is related. Let's say you run a website selling gardening tools. You write an article about the importance of weeding around seedlings when planting a native plant garden. It's natural, in this case, to link from your page about weeding, to your page selling a stirrup hoe, which is a tool for weeding. Look at the total picture of your website content and find the relationships between topics. Linking to highlight these relationships is meant to improve user experience on your website, letting a visitor know that if they are reading the page about weeding in native plant gardens, your stirrup hoe is a really helpful tool for this. By finding the relationships, you will be enriching the navigational architecture of your website.
Increasing links pointing to your site from external sources typically revolves around two things. The first scenario relates to links you earn based on merit, without having to ask for them. Let's say you write the best article on the web about planting native California roses in the garden. You may find that the usefulness of the article earns links on its own, perhaps from garden enthusiast organizations, gardening blogs, native plant societies, and even government sites talking about water conservation through native gardening. Over time, if your article is truly useful, it may earn some very good links from third parties.
The second scenario involves actively asking for links. Let's say you're going to give a talk at a local university about native plant gardening. You could ask the university to link to your website from their upcoming events page. You could ask the local newspaper to do the same. You could ask local gardening clubs to do the same from their websites. In fact, you could create a piece of static content that summarizes your presentation and offer it to third parties and a resource that might be of help to their readers. You could earn some links from that, if what you create is good enough. I'm emphasizing this, because unless you have something of note to publicize, it's not going to be easy to get people to link to you. So, merit is still involved here, but the difference is that you are actively asking people to notice what you are doing, whether that's an event you're participating in, a video you've created, an eBook you've published, an article you've written, a graphic you've created, or what have you.
On a final note, if your business model is local (like a retail store), you'll also be building some links on local business directories like YP.com, Foursquare, Bing, etc. You'll be creating business listings on these platforms that link to your website. But if your business model is virtual (like an e-commerce site), then building links on most directories is typically a waste of your time.
If linkbuilding is new to you, please do check out the section on this exact topic in our Beginner's Guide to SEO: https://moz.com/beginners-guide-to-seo/growing-popularity-and-links
Hope this helps!