Do not write your own library from scratch. It sounds a whole lot easier than it is since most of the work in writing the library isn't the actual code writing but rather getting it tested and debugged in all the variants of all the browsers. Use a library.
As for which library, they provide mostly the same functionalty at the bottom but differ in the high level feature set and how they're organized. My take:
* jQuery - Most popular, best for 'ajax sprinkes' apps but less great for full-on applications
* prototype - Prototype hacking outside of ES5 feature backfilling is evil
* mootools - The community cares a lot about their visuals but I got burned with a number of bugs caused by them taking shortcuts for size. This was 4 years ago so the kinks are probably worked out, but there are other options so meh.
* yui3 - My favorite. Built around a selector api, fine grained module system, custom event based, and the UI layer is very much about progressive enhancement. The downside is that there's not a lot of community support.
* sproutcore - the absolute best data layer, UI layer wants to own the page
* dojo/yui2/closure - I don't like these because I think they're too heavy or too much like java
My specific problem with jQuery is that every app over 1k lines or so tends to go spaghetti. The module/event systems in dojo and yui and the data binding in sproutcore guides you to a more modular system which makes for a more pleasant application building experience. There isn't really a technical reason you couldn't do it with jQuery but I've never run across an elegant, modular jQuery codebase. If anybody has one, I'd love to see how it's structured.