Hard question to answer without knowing more about your photographic skillset, budget, and what you want to achieve in the long term.
A point and shoot will give you nice shots (a phone camera is just as good), but might be a bit restricting if you want to learn more about how photography works
A 'bridge' camera is the next step towards full control - you'll have more zoom to control the framing of your shot, and more control over the configuration of how the photo is taken, but not removable lenses etc
Mirrorless cameras are a sub-set of DSLRs, they don't tend to have a viewfinder, you're looking at an LCD screen to help you create your photo, but other than that they are DSLRs with removable lenses.
Full DSLRs (Digital Single Lens Reflex) give you all the control, but will cost a lot more, especially once you get into it! They come in 2 varieties, APS and Full-size sensors (the difference is the size of the sensor capturing the image - essentially similar to different sized films), and there are several different lens-fitment systems: Nikon/Canon/Pentax/Olympus etc etc are all different, but you can use the vast majority of (say) Nikon lenses on any given Nikon camera body.
I personally like a simple, but controlled experience so I have a Nikon D50 body* with a standard 'kit' lens** and a couple of older, fully manual*** lenses which, conversely, are better quality glass than the Kit lens, despite being older.****.
What you want to buy will depend hugely on what you want to achieve, but if you're definitely thinking about going down the DSLR route, do a good amount of research because once you buy (say) a Nikon, you open the doors to buying more lenses (always Nikon fitment), and if you suddenly decide you want a Canon body for some reason, you'll have to replace ALL your existing lenses - which can get expensive!
* Nikon D50 - was a cheap, bottom of the range, but very capable at the time, 'pro-sumer' body with a 6Mp sensor and a cut-down suite of presets and control options compared to a fully professional body (eg Nikon D100).
** Kit lens - the lens the manufacturer chose to sell the body with as a 'kit' for consumers.
*** Fully manual - no auto-focus, no chip in the lense to identify it to the camera body and allow the body to control the lens aperture, so aperture is controlled manually on the lens, shutter speed by the control on the camera, and metering (balancing the exposure by control of aperture/shutter-speed/ISO) is by educated guesswork.
**** Older lenses tend (not always) to be made somewhat stronger than modern plastic lenses, this is a generalisation. They're available in very good quality, second hand, from all the usual sources.