OK...here comes a big fat bunch of numbers - follow me if you can, as I'll try to present a lot of info but hopefully in a way that can be followed in the end. I'll stick with the well-used '35mm equivalent' number as it is the most commonly used.
Those '35mm' equivalents are based on what is commonly referred to as a 'full frame sensor', based on the 35mm film standard. What happens whenever you go with a sensor that's smaller than the big full-frame sensor is that you start to introduce a 'crop factor' - that is, because of the smaller sensor, you are seeing a smaller section of the full frame - the effect would be similar to enlarging the full frame view on your monitor to 'crop' into the photo - you are 'zooming in' on the photo which crops out the edges of the frame, presenting you with a smaller section of the photo to view. Most DSLRs are what's known as an 'APS-C' sensor...these sensors have a crop factor equivalent of 1.5x to 1.6x. Which means that a lens labeled 24mm on an APS-C camera provides the equivalent field of view of 24mm x 1.5, or 36mm. P&S cameras have even more 'crop factor'...the sensors in compact cameras are itty-bitty little things...most P&S cameras have a crop factor of around 6x (varies a little by camera). The lenses have such small numbers on them that P&S companies don't even bother to tell you those numbers - they prefer to just give you the 'zoom' amount, and if you dig for more info, they'll tell you the '35mm equivalent' lens range, which has already factored in that 6x crop. So when you have a P&S camera with a '36mm equivalent lens', what that camera really has is a tiny 6mm lens and a 6x crop factor. Look on the lens of any P&S camera - you'll usually see tiny little numbers like '3.5-4.6/6.18-24.7'. That's all the true lens info right there - the first numbers are the maximum apertures (F3.5 at the wide end, F4.6 at the telephoto end), and the true lens focal range (6.18mm x 24.7mm). Factor in the crop of the tiny sensor, and you've got a 37.08mm x 148.2mm range in 35mm equivalence.
How much 'x' a zoom has is always based on what the widest lens setting is. The bigger number of 'x' DOESN'T always mean you can see distant objects better. If a camera has a wide lens setting of 24mm, and a 12x lens, you would multiply the 24mm by the zoom factor of 12, and you get 288mm. Now, take a camera that has a 36mm wide setting, and say that camera only has a 10x zoom. Less than the 12x zoom, right? No...because multiplying 36mm by the 10x zoom factor, you get 360mm. The 10x can see farther than the 12x! But the 12x camera can fit more into the picture when standing up close, because it can see wider.
When going to DSLRs, they stop using the 'x' figure, and just stick with those 'mm' numbers. You can see right away that most DSLRs you buy are APS-C sized sensors, and remember they all have a little crop factor (Canon's APS-C cameras are 1.6x, Nikon, Sony, and Pentax are all 1.5x...I'm sticking with the 1.5x for ease!). So when you get a kit lens that's an '18-55mm', you multiply your crop factor of 1.5, and you end up with a 35mm-equivalent range of 27mm to 82.5mm. How many 'x' the lens is can be determined by dividing the telephoto by the wide, or 82.5 divided by 27...you get roughly 3x.
You can see why superzoom cameras are popular as all-in-one travel cameras - your S5 with the 36mm wide equivalent and 12x zoom has a telephoto equivalent of 432mm. To get that same range in one lens on a DSLR would cost...nope, nothing available in that range! But all is not lost. As mentioned, you CAN get some DSLR lenses that function much like a superzoom camera - the afformentioned Tamron 18-250mm lens, for example, is nearly 14x, and on most DSLRs provides the equivalent of 27mm to 375mm. Considering it can go wider than your S5, though not as long, it has the same convenience factor. And there's the sensor size and image quality to consider - at the resolution levels of most DSLR cameras today, from their massively larger sensors compared to your P&S camera, you can crop your photo as much as in half, and still have the quality you had with your P&S camera.
Hope all of that wasn't too confusing. When you decide to get your DSLR - and make sure you are ready to go to a bigger, heavier camera with extra lenses and such - for all the wonderful benefits you can get from it - you might shop around a bit and consider getting just a camera body with no kit lenses (if available, you can save a little by opting for no kit lenses) and buy one of the 'superzoom' lenses like an 18-200 or 18-250 right off the bat. That way, you'll be starting with the DSLR with one nice, convenient lens that you won't have to swap or change, and it will give you the same type of convenience you had with your S5. It will be a little pricier - but so much more convenient, and you won't have to look for a 55-200 lens later. Some folks even find one lens like this is all they ever need.
One more tip too - don't be afraid to buy used lenses - you can find some great older lenses in your camera mount that will work very well and save a ton. Best of luck!