I am thinking about purchasing a Canon 60D and selling my Rebel XT and wondered if I would need to purchase new lenses? I only have two but as you all know, once you invest it is hard to not take those into consideration when purchasing a new camera.
The EF lenses will work on any modern Canon EOS camera (including both the XT and the 60D).
The EF-S lenses will work on some digital Canon EOS cameras, including the Rebel series, XXD series (including the 60D) and the 7D. You can see a full list of supported cameras on wikipedia's EF-S page.
EF-S lenses do not work on the 5D or 1D series of Canon EOS DSLRs, nor do they work on the D30, D60 or 10D which are three of Canon's earliest DSLRs (not to be confused with the 30D or 60D, which both support EF-S lenses!).
So to answer your question directly, yes, any lens that works on your Rebel XT will work on a Canon EOS 60D, since both support the exact same set of lenses (EF and EF-S only).
Just in case the question comes up, Canon FD lenses are NOT interchangeable with the current Canon line. The FD is the previous generation, so...
Also, for a complete list of cameras that support the EF-s mount, see the wikipedia page. Of note is that the 10D is not supported, or similarly old cameras.
In addition to the other good answers - You did not mention what lenses you currently have. Thus, my guess is that these are the (18-55mm) kit lens and the 75-300mm tele. This is a very popular combination with Rebel owners, including me. Once you consider changing your camera body to the next step in the line (xxD), you may also consider that relative to that camera's price, these lenses are relatively cheap, and together cost about $250 (which most of it can be even returned if you sell as used). So, upgrading from a Rebel may be a good time (maybe your last chance) to consider if lens compatibility really is an issue here.
In the particular case of the EOS 60D this is not really an issue, but if you ever considered a full frame body (EOS 5DmkII) or even converting to other brand, than lens compatibility should not hold you from doing this.
There are four Canon lens mounts in active production.
EF: Lenses made for the EF mount are compatible with all of Canon's film and digital EOS SLR cameras, the first of which was released in 1987.
EF-S: Lenses with this mount are compatible with Canon DSLR cameras with APS-C sensors (####D, ###D, ##D, 7D). Canon lenses with this mount have an additional component that protrudes from the back of the lens to prevent it from being mounted on full-frame bodies. Third-party lenses made for APS-C sensors typically lack this component. A few very old Canon DSLR bodies with APS-C sensor are incompatible (D30, D60, 10D)
EF-M: This is a new mount released in 2012. It is compatible with Canon mirrorless cameras with APS-C sensors (EOS-M). There are adapters that allow EF/EF-S lenses to be used with EF-M bodies.
RF: This is Canon's latest mount, announced in 2018. It is compatible with Canon mirroress cameras with full-frame sensors (EOS-R). There are adapters that allow EF lenses to be used with RF bodies.
Older mounts, which are no longer being produced, are primarily of interest because they may appear in search results for "Canon lenses".