What equipment could I use for digging out a basement after the house is built?
The standard (as much as there is one) technique for this these days is to jack the house up to the point that smaller power equipment (which is not cheap) such as a skid-steer loader can be used - and so that the basement walls can be removed (with the house supported on blocking) and replaced with new ones to full depth, properly supported on footings. Some types of "trencher" may also be of use in this job. Sometimes the house is simply rolled off to the side once jacked up so that a new foundation can be built (without the house in the way), or a new foundation is built on the property and the house is moved to it, then the old one is demolished.

There are companies you can hire for the job, and that is by far the most usual approach to it.

Unless the current foundation is "full-depth" (ie, walls in deep trenches, with the dirt left on the inside for later digging out) it can, as suggested by @djs be hazardous to both your house and your life to dig below the current level of the footings that the walls (and house) sit on.

One obvious improvement from your described approach would be to build a ramp from the basement (possibly involving digging some outside the house) so that you can use a wheelbarrow or cart to roll excavated dirt out from under the house, rather than carrying it in buckets. In some cases a conveyor might be of use, but that's a fairly expensive and fiddly (lots of moving parts, and sensitive to overloading, etc) approach. If you cannot make a long ramp and have a typical steep bulkhead, wheelbarrowing to the bottom of the bulkhead and transferring to a smallish box on wheels running up a track (probably wooden) on the bulkhead that you pull up from outside (thus smallish- too heavy makes it miserable) would be much better than carrying buckets upstairs.

In the more desperate case where you don't even have a bulkhead, a ramp could be built to allow you to roll a box of dirt out a window, if there are at least basement windows to work with.