I changed the batteries and I replaced the remote. However, when I put it on time delay and press the button nothing happens. Is this a remote issue or a camera issue?
The Rebel XT has a single 10 second self-timer setting. You should set the camera to Self Timer using the Drive Mode button on the back of the camera. The self-timer mode will stay active for four minutes. If you don't use the remote during the four minute interval, the self-timer mode will be cancelled.
Even though the self-timer mode includes a 10-second delay when activating it with the camera's shutter button, with the RC-1 infrared remote the photo will be taken either immediately or after a two second delay when you press the round button on the RC-1, depending on the position of the switch on the RC-1. The dot is for immediate release of the shutter, the "2" is for a two-second delay. This can be useful if you don't want the remote to be visible in the picture. The "L" is to lock the remote so accidentally pressing and holding the button down won't drain the battery.
It's covered on page 49 of the EOS Rebel XT/350D Instruction Manual
Here's link to a copy of the RC-1 Instructions
Without more information from you, it's hard to say where the problem may lie. Here are a few things to look at.
The RC-1 is an infrared remote. Much like a TV remote, it has to be pointed directly at the sensor on the camera. The sensor is that white dot on the front of the grip of your camera body. If you're pointing the remote at the back of the camera, or if you're shooting selfies outside in bright sunlight, the signal from the remote may not be reaching the camera's remote sensor.
If you need to shoot from behind the camera, consider a cable release instead. If you need to shoot from farther away than 5-10 feet, or you need to shoot in bright sunlight outdoors, then consider getting a set of cheap radio flash triggers that can double as a shutter remote.
The RC-1 runs off two CR-1220 button batteries. If either battery is bad, or you can't tell how old the ones you purchased were, this could be an issue. If you can test the batteries, or know you got ones that were in good health, then you can eliminate this as a problem.
You'll also want to check that the battery in the camera body is good.
Drive mode on the camera
The 350D must be set to the "self timer/remote control" drive mode, otherwise it's going to assume you're going to use the shutter button on the camera body to release the shutter. Make sure the little timer/remote icon is showing in the LCD panel (the one on top of the monitor panel).
Switch on the remote
The RC-1 remote has a three-position switch. The positions from top to bottom are are 2 (2-second delay), • (shutter release), L (locked). Check that the switch is not in L, and that you have it set to 2 if you want a 2-second delay before release.
Just like when you're shooting with your finger on the shutter button, the shutter will not release in any mode that requires achieving autofocus lock if that lock has not be achieved. Low light, the lens being in manual focus, or shooting inside the minimum focus distance of the lens can all be issues here. A half-press on the remote's button should work the same as a half-press on the camera's shutter button to set AF. Check and see if you can see/hear your camera attempting and failing to autofocus. If you're doing selfies, consider setting up a dummy target, and manually focusing on it before getting in front of the lens.