...It really depends on what you want in your game. First off, pre-made game engines such Unreal Engine, Unity, RPGmaker, etc, have been developed by very talented programmers who have been working on them for many years. If you are looking for a pre-made, professional written engine; it will almost always be better for you to obtain licenses rather than take the time to make your own (Mass Effect, Hawken, and Far Cry use pre-made engines). It should be stated that many game engines such as UDK or Unity will allow you to utilize some form of their engine for free until you start making profit, in which case licensing will be required to do some licensing, but these engines require far less time to develop games for. It should be noted that there is no 'best' engine; each and every single one will be written and designed differently and have various strengths and weakness's. Do research before committing to one...
...The other option for creating your engine is to simply write one from scratch. Many indie as well as top AAA developers have used proprietary (custom made engines owned only by the devs) specifically built for a single game project (ie, Minecraft, Skyrim, Cube World, etc). The advantage of this method is that your engine can run your game exactly the way you want it (in some cases, this is far better than using pre-made engines); given of course that you engineer it just for that game. The disadvantage though is that an engine is only as good as the programmer who writes it, and it will take time to make a prototype version. However, as the creator of the engine, you will not need to make any deals or licensing agreements to utilize it...
...As far as budget goes, the only particular expensive aspects of an engine are the lighting, AI, physics, and particles. In the past, game makers simply changed the colors of their textures to make fake lighting, implemented very basic "physics", and scripted AI behavior. Today however, there is much, much more involvement in these features such as light-maps, dynamic AI, ultra realistic particle effects, etc. Although you can write that into your engines, it is often difficult for many game developers to do it effectively on larger projects and instead opt to pay licensing fees to use pre-made plugins (ie, Havok, Nvida Physx, etc). Often times, the plugins are more complicated then the actual engine, especially the AI...
...Also, here is an example outline of a game development cycle:
1. Plan plan plan, this cannot be overemphasized. If you don't get this down right, you will be in big trouble.
2. Develop a working prototype engine, and build up from there. Be sure to fix bugs before adding new features or optimizing.
3. Start creating game content and mechanics as you outlined in your plan.
...
...There are also guides on the internet; they are on youtube channels, engine fourms, blogs, etc. There are also very detailed books in libraries. Some colleges offer free or cheap game development boot camps. If you are serious about your project, you can always consult them...