We have a number of good modern reactor designs like the Integral Fast Reactor, but they're not perfect. More research needs to be done to develop even better reactor designs that produce less low-level radioactive waste.
The IFR itself is a 12 year old dead project, but the same concepts are being worked on again in the form of Advanced Burner Reactors. For more detailed information on modern reactor designs, see Wikipedia: Generation IV Fast Reactors.
With proper nuclear fuel cycles and efficient use of the world's uranium and thorium reserves, nuclear fission should be able to meet the world's energy needs for over ten thousand years. That should be more than enough time to develop replacement technology.
The real reason not to worry about what will happen when we run out of Uranium in a few thousand years is Nuclear Fusion, which promises nearly unlimited clean energy once we get a viable design.
Most of the effort in international fusion research is based on a design called the Tokamak, which is large, expensive, and is expected to be able to produce massive amounts of commercial electricity by 2080. That date isn't just an arbitrary "far future" date, but a realistic estimate based on current experimental plans. Even if that's off by a factor of two, we still won't have run out of coal, much less Uranium and Thorium.
The focus on the Tokamak design has, unfortunately, had the side effect of deflecting research money away from other efforts which have the possibility of producing viable fusion power generation much sooner. For example, Dr. Robert Bussard - one of the founders of the United States Tokamak program - claims to have developed a fusion power design called the Polywell that could be commercialized inside of five years. Even if that specific design has some non-obvious flaw, general fusion research is significantly underfunded compared to the advantages a breakthrough in the area would bring.