One of my switches is being replaced. It is a 24 port 100 base MTRJ Cisco switch. It is being replaced with a WS-C3750X-24S-E. Will this switch support 100/1000 or 1000 base only? I have some devices that do not support 1000 yet. If it does support 100/1000 does this depend on the SFP's? I thought it would auto negotiate the speeds but I am being told that it isn't.
You don't say which switch you are replacing nor specify the type of transceiver it has (you only specify the connector). However, the only 24 port Cisco switches I know of personally that have 100Mbps MT-RJ ports utilize 100base-FX, but make sure you double check.
Will this switch support 100/1000 or 1000 base only?
According to the Cisco 100-Megabit Ethernet SFP Modules Compatibility Matrix, the WS-C3750X-24S can utilize GLC-GE-100FX transceivers so it can operate at either speed. This should work fine for your 100Mbps devices provided the assumption I made above is correct.
If it does support 100/1000 does this depend on the SFP's?
If you use Cisco transceivers, then yes it does depend on the SFPs. AFAIK, all Cisco transceivers operate at only one speed or the other.
Some third party vendors do sell transceivers that operate at either 100Mbps or 1000Mbps. Keep in mind that these are not officially supported by Cisco and you will have to do your research to make sure they will work in your switch.
Also, since it seems you are somewhat new to fiber, make sure you are using compatible transceivers. In your case, if your existing devices are using 100base-FX, you would not be able to use a third party 100/1000base-SX transceiver.