Well, I'll be both the voice of reason to some degree, and also the main Disney Villain for this thread (oh, I will be hated! Enmity will ensue!).
First off, the reason - I'm not sure exactly which P&S Sony you have, since they have 20+ models, and they vary greatly - from ultra-compacts to standard compacts to slimlines to ultrazooms. There is not much uniformity between them just because they are by Sony - some are ultra-fast, some slow; some have excellent glass, others average; some handle low light very well, some stink in it.
But there are things you can do to improve the performance of your P&S camera, even if it is one of the lesser models. First off, as you mentioned trying - half-pressing the shutter will definately speed up the operation. Fully pressing a shutter in a P&S camera forces the camera to: meter the scene, decide on an aperture, ISO, and shutter speed to use, approximate white balance, focus, and shoot. That's alot of decisions the camera has to make all from one press of the shutter - which results in a lag. But if you half-press first...by anticipating the shot you want to take, you force the camera to do everything except take the picture...so when you fully press the shutter the rest of the way, you get near-instant response. It's easy...and in fact is what nearly every photographer does. If your child is getting ready to blow out candles on a cake...half press on their face while they are making their wish - the camera will lock metering and focus and wait for your final command. Then as they lean forward to blow, you snap the shot. That's all it requires - a little planning before you shoot.
Now comes the part where I become the board villian. Sony makes excellent cameras. (Tomatoes flying, pitchforks and torches approaching from the masses!) I know it is something people are not really allowed to say, and 95% of people who come to camera boards are long-time photographers, who were shooting with film too and who live and die by the name 'Canon', and a few 'Nikons' thrown in to boot. And when it comes down to the ultimate digital photographic tools, Canon and Nikon DSLRs are certainly about the best you can get.
However, not everyone will be able to get great performances out of a DSLR - having one doesn't make you a great photographer, no more than owning a Porsche will make you a great driver! And for a person who really is not a Photographer with a capital P, using one and getting good results from one can be more challenging sometimes than using a good point and shoot.
How many digital camera amateurs look at the LCD screen on their cameras, and can see what the shot is going to look like without having any understanding of the settings? How many just move the camera around, watching the LCD, until the picture looks pretty good - not really knowing how all the automatic features are doing what they do? Well if they go to a DSLR, they won't be able to do that anymore. No live preview means you look through the lens, but can't see the effect of the metering, shutter speed, white balance, etc until AFTER you've taken the picture. To a skilled photographer, this is preferred - TTL gives much more accurate ability to manually focus a camera especially in low light. But then again, they know what the shutter speed and aperture chosen are going to produce for their shot - and they would know if they need to dial in a little EV or raise the ISO sensitivity even without seeing it. Most basic amateur P&S users wouldn't...and often get pretty bad results from their new DSLRs that they felt compelled to buy!
How many amateur photographers go into advanced editing suites and alter their photos to fine-tune them for maximum results? Not many - most just post what they shoot. When people look at the marvelous results pouring out of DSLRs by excellent photographers - they think they can just go buy one off the shelf with a kit lens, take the shot, and post it direct from camera and see those same astounding results. Well - they can't! These wonderful DSLR users with their excellent photos don't use kit lenses - they use very expensive lenses that often cost several times what the camera body alone costs. And they almost always shoot in RAW format, which is then loaded onto a computer and loaded into a RAW converter software where the picture is tuned for color, contrast, sharpness, hue, tone, brightness, saturation, shadow recovery, highlight recovery, and more. And it is edited in Photoshop to get rid of unwanted elements, run through noise-reduction software to get rid of the graininess, run through high-pass or USM sharpening to get the edges just-so. Too many people get talked into going the DSLR route who really didn't want any of that, and find themselves taking pictures in Auto mode set to JPEG output, and getting results no better, or even worse, than with a good P&S.
Also, when it comes to P&S cameras, the sensors, the software internally, and the processing algorithms mean as much to the output as the lenses...these cameras are tuned to be as effortless as possible for less skilled photographers, but still grant some additional control for those who know what they are doing. Canon and Nikon are skilled mechanical camera makers, and lens makers...but hold no advantages over Sony in the electronics and technology sectors. In fact, virtually every Canon and Nikon P&S camera uses the exact same sensors in your Sony P&S - which are MADE by Sony. Look inside a Canon SD-series, or S-IS model...and you'll find a Sony sensor. As for lenses, Sony doesn't know much about them - which is why they contracted legendary German lens manufacturer Carl Zeiss to design and test the lenses on most of their cameras. With excellent features that most other P&S manufacturers don't have in their cameras (live shooting histograms, infolithium batteries with minutes remaining readouts on screen), Sony P&S cameras are at the top of the heap - at the most, other manufacturers' cameras are equal to the best Sony models - not better than.
Personally, in avoiding the usual Canon bias that you will find on nearly every camera discussion board on the internet - I would recommend certain P&S models from Sony, Canon, Panasonic, and Fuji...each with their own strengths. Sony's H-series cams are well featured, built well, have good lenses, and have very good image quality overall. Canon's S-IS cams have good lenses, very good image quality, and good reliability reputation. Panasonic FZ cams have probably the best lenses overall, with excellent optical quality and good IQ, but not as good at higher ISOs with a fair amount of noise. Fuji's F20-50 cams don't have as good of lenses as the others, and IQ is good but just a touch below Canon and Sony...but they have by far the best low-noise, low light sensor - blowing Canon, Sony, and Panasonic out of the water.
When it comes to DSLRs, Canon is probably the best overall, with Nikon just a notch down...then Sony, Olympus, Pentax, and Panny another step below. But in the P&S market, Sony is right at the top, and has some very competitive cameras, awesome speeds, lots of features, great glass, and good prices. And yes...in the interest of disclosure, my galleries were taken with a Sony P&S (booo, hisss...I know!). But I've also had Canon, Minolta, and Pentax cams before - and still have my Canon SLR kit with a few lenses. Someday I may decide to get a digital body - but personally I really like P&S digital cameras...they offer me something that a DSLR cannot. For under $500, I have a small, light camera with an optical range of 36mm to 432mm, good auto and program modes plus some manual controllability when I want it, decent all-around performance in any situation, and all in a single, sealed body...no bag-o-lenses, no 2Lbs of weight hanging from my neck...just portable freedom to enjoy myself and snap a pic when the whim hits me, and still get a wall-hangable shot!
And now I shall be lynched for actually liking a P&S, not feeling like I need a DSLR just now, and for *gasp* supporting a Sony product!
;D 