I think if you want to switch from a beginner DSLR to mirrorless there are plenty of options. If you want to switch AND UPGRADE, there are only two at the moment: Sony a7 and FUJI X lines. (Three if we add Leica)
Both have great image quality. Sony a7 has the advantage of full frame, FUJI has superior lenses.
All you need is to decide if you really need a full frame system, what lenses are you going to buy for your camera and how much will it all cost. Both systems, if we talk about a camera plus 4-5-6 lenses, are not cheap.
I have never tried X-T10, but I have X-T1 which is similar. The image quality is exiting compared to beginners DSLRs, and the "kit" 18-55 lense is so good, it is just ridiculous.
35 mm is a great, great lense, highly recommended, 56 mm is top class, even small pancake 27 mm, if you want it light and compact for street shooting, is very good.
55-200 is a bit slow, but excellent as well and quite compact for its IQ. 10-24 is a very competent lense as well. I have switched from Nikon APS-C system and was amazed by FUJI XF lenses quality.
I do not know about Canon, but with Nikon I always felt (I do not state that it is indeed the case) like their APS-C lenses were intentionally dumbed down to encourage you to switch to FF lenses that are superb, but big, heavy and will cost you an arm and a leg. And a shoulder, if you carry it around all the time. FUJI do not produce FF systems , so they concentrate on top quality APS-C line that can compete with the best of Canikon.
You need to consider how much you will spend on lenses and whether the price difference between XT-1 and XT-10 is relevant is the context of the cost of your future system. XT-1 has one clear advantage in a great, huge EVF which is fun to use. It is just like looking into a big TV screen with all info you need, compared to a tiny baby DSLR OVF.
If you go for a mirrorless, you will need some spare batteries, a mirrorless is much more power hungry than a DSLR. I use cheap third party batteries without any problems, you can get a couple for $15 , they work similar to the original ones.
WHOA! Thank you for the incredibly detailed and precise response of your answer. You have actually found the crux of the problem and I think I just made up my mind about what system to go for.
Sony has the superior camera (for the price) and a wider range of lenses but with Fuji having the better quality lenses (especially with such a good kit lens), I am SOLD on the Fuji. Someone else in the thread (forgetting who) reminded me I look long term and shop for lenses rather than for cameras so that's what I'll do: either a new Fuji X-T10 or a used/refurbished Fuji X-T1 will be what I'm getting I think.
Thanks.
One last bit of advice before you commit.
FUJI is quite special due to its unorthodox non-Beyer array X-Trans sensor, which some love and others are quite critical of. I think it is down to a personal liking/perception of color, detail, sharpness and other such things, that this sensor renders a little bit differently. It is hard to argue that FUJI approach is special. But "special" is not necessary best or even better. Some do not like it.
I personally have switched to FUJI because I LOVE FUJI colors. This is a company with a rich heritage going back to the film era when FUJIFILM had to compete with Kodak on the 35 mm film market. To compete they had to create their own different but coherent "color philosophy" and they have successfully transferred it to the digital era.
FUJI is also known for its great skin tones and spot on white balance if you shoot JPEGs. I often shoot JPEGs and sometimes find it difficult to match when converting RAW files. But not all RAW convertors do the best job with X-Trans files. If you are using Adobe you might want to reconsider, although they are becoming much better with X-Trans. I prefer Capture One Pro, it is excellent.
To sum up, FUJI is a system with a signature. Some love it, some prefer a neutral approach.
I would advice you to browse the internet, sites like X-photographers to see if you like the end result.
Amongst the reviewers I like Mansurov, he is a good pro and quite unbiased as far as I can see.
Fuji X-T1 Review