If you're using TTL flash with Av mode all Canon EOS cameras, including your Rebel T6i/750D, will allow you to set the shutter time to either: 30" ←→ 1/x-sync (Auto), 1/60 ←→ 1/x-sync (Auto), or 1/x-sync (fixed) where 'x-sync' is the denominator of your camera's flash sync speed. In the case of the EOS Rebel T6i/750D, the flash sync speed is 1/200. How to select the above options is covered on page 174 of the EOS Rebel T6i/750D Instruction Manual.
Most Canon cameras above the entry level Rebel/xx0D and Rebel/xx00D line include a capability called 'Safety Shift' that may, depending on the model, allow you to choose any selectable Tv as a minimum Tv when using 'P' (Program AE) or 'Av' (Aperture Priority AE) modes. This answer discusses how to combine Safety Shift and TTL flash in low light, but some of it can also be used with a camera that does not offer Safety Shift.
- Set the aperture as wide as is available that you can live with in terms of depth of field.
- Set the flash to E-TTL and dial in between -1 and -2 stops flash exposure compensation. Take a few sample shots and dial it in based on the results. Be sure to match the camera's LCD brightness to the ambient lighting. Double check exposure using the histogram.
- Make sure the menu option for shutter speed with flash in Av mode (for your EOS Rebel T6i/750D it is at: [Shooting (Red) Menu Tab 1] → [Flash Control] → [Flash sync speed in Av mode]) is set to '1/200-1/60sec. auto'. Essentially, this will manually set your Tv to 1/60 second for most of your shots using Av exposure mode in dim light. When you are pointed at a "hot spot", or the lighting overall is brighter, the camera will reduce the shutter time to as short as 1/200 second (your camera's flash sync speed). You can also select '1/200 sec. (fixed)' if you want to lock the shutter speed at the flash sync speed. Any automatic variation in exposure of ambient light would then require 'Auto ISO'.
- Set the ISO to whatever you need to allow the ambient light to show the background a stop or two underexposed. With the Rebel T6i/750D and other EOS xx0D/xx00D models, if you are in M exposure mode, you can not enter an (ambient light) main 'Exposure Compensation' value for affecting 'Auto ISO'.² You can still enter a 'Flash Exposure Compensation' value for affecting the E-TTL computed flash output.
- If the lighting is mostly tungsten (warm), then consider gelling your flash with orange to match the output of the flash to the color of the ambient lighting. If the primary lighting is fluorescent, then use a greenish gel. For other types of light sources, use an appropriate color to match your flash to the predominate ambient light.
On the other hand, if you're shooting in 'P' (Program) exposure mode with or without flash, the default setting for all Canon EOS cameras is a shutter time no longer (slower) than 1/60. With a dedicated flash¹ active, the shortest shutter time allowed will be the flash sync speed. If no flash is detected by the camera, the shortest shutter allowed with be the camera's minimum (shortest) shutter speed - 1/4000 in your case. With 'P' mode in dim light the camera will tend to use 1/60, raise ISO up to ISO 400 (if 'Auto ISO' is selected), open up the aperture all of the way, then raise the ISO even further if you have ISO set to 'Auto'. You can set both main 'Exposure Compensation' that will affect the exposure of the ambient light using 'Auto ISO' (if you're already at 1/60 and the lens' maximum Av) and 'Flash Exposure Compensation' that will affect the calculated E-TTL flash output power in P exposure mode.
When in not ideal lighting conditions I usually target 3200 at most.
ISO 3200 is still pretty dark with an f/2.8 lens in dark environments. Sometimes you just have to kick the ISO up and live with the noise to get any kind of usable shot. If you are getting 1/25 at ISO 3200 and f/2.8, even an f/2 lens would only get you to 1/50. You'd have to shoot at f/1.4 to get 1/100 at ISO 3200 which, in such dim light, would probably mean your AF would not be accurate enough for the razor thin DoF.
When I'm shooting in such environments without flash, I tend to set all exposure parameters manually for what is needed when the varying lights are at or near their brightest. I then compose the scene and "sit on it", sometimes with the shutter half-pressed or the 'AF-ON' button pressed, until the lights match the exposure I have set and then take the picture. It takes a lot of patience, as the moment when you'd like to time your subjects may not coincide with when the lights are where you need them.
The other option is to add light of your own, preferably from off the lens' axis. If your environment allows it, bounce the flash. If bouncing is not an option, even an off-shoe cord and a flash bracket (or holding the flash wide with your left hand) will give better results than mounting the flash directly to the camera's hot shoe. A wireless trigger and a remote flash is even better.
For more about how I approach shooting with or without flash in the kinds of lighting you seem to be talking about, please see this answer to Nightclubs photography, setup to capture the real mood and atmosphere?
Please also see:
How can I take photos during a school dance?
First time - Night Club Photography - what gear do I need?
What should my plan be for photographing in a nightclub setting?
For an even more general question about shooting clubs, bands, concerts, theater, etc. with a lot of links to other existing related questions here at Photography.SE, please see: Best ways of photographing at a concert/festival
You might also find Blown out blue/red light making photos look out of focus helpful. It talks about how exposing to not blow out the dominant color channel and how vital post processing the raw data is for images shot under poor lighting such as that often found in clubs.
¹ 'Dedicated' flash would be any flash that the camera can detect is active, whether it is set to E-TTL or Manual flash power. In general, this would be any flash, Canon or third party, that has a full set of contact pins in the Canon pattern on the hot foot. Some manual only flashes with only a center pin will not be detected by the camera, even though they will receive the signal to "fire" when the shutter is opened.
² Many higher tier Canon EOS bodies do allow setting exposure compensation in M exposure mode to affect 'Auto ISO'.