Assuming strings have been bedded in, and all are properly in tune to standard, then the whole guitar is in balance.
That is, the strings exert a tension against the neck and the vibrato system, usually springs.
When one string is changed, that balance is changed.
Lowering a string pitch will loosen that string, so the opposing part of the balance becomes relatively stronger.
That then exerts more tension on the remaining strings, putting (pulling) them out of tune.
The same happens when a string breaks.
The tension balance is changed, so making the other strings out of tune.
Any guitar player should be capable of changing strings on the guitar he plays, and subsequently putting that guitar in tune.
On some with a floating vibrato it will take some time, but that's the nature of the beast.
When new strings are put on, unless they are the same as the old set, other changes may need to be made: neck tension, action, intonation may all need tweaking to accommodate a different set of strings.
At this point a lot of guitarists will turn to an expert.
However, I don't think those are your causes.
It's merely the change in balance caused by the change of tension.
Handy hints - learn to change your own strings, get used to tuning the whole guitar rather than one string, don't just rely on a tuner to tune one string.
If we think our guitar is out of tune, chances are it was our ears that told us !
Or it might have been the drummer!
So our ears should be good to put it back in tune !
music.stackexchange.com/users/2808/Tim
If the strings are nylon then they will require many days of constant retuning before they stabilise
music.stackexchange.com/users/13752/Tony Schaps
If it has a tremolo that is not set flush to the guitar body, then retuning one will affect the tuning of the others, by altering the overall tension against the springs.
music.stackexchange.com/users/1344/luser droog
Is the guitar in tune with itself(ie 5th fret low E string ==
A string etc)?
If this is the case the problem may be with your tuner.
music.stackexchange.com/users/6928/david strachan
Aside from the "Drop-D" issue, definitely DO get into the habit of doing your own string replacement and tuning.
Besides gaining the experience, even the best guitars need re-tuning several times with new strings until they settle in and stretch out.
You can help this along by changing one string at a time, so the neck tension is only changing a little at a time.
This also gives you an easy tuning reference that you can hear, less dependent on the electronic tuner.
Remember, electronic tuners are most useful when the string is NEAR its proper pitch.
They aren't very helpful and are even confusing until you at least get the string approximately tuned, and even then your EAR has to eventually become the final reference.
That is because guiar tuners are near perfect, but most guiars (especially beginner guitars) are NOT.
Then, each time you get one of the new string approximately tuned, bend the note
(I sometimes grab the whole string up near the picking area with my thumb and first finger and give it a sharp twist).
The pitch of the new string will drop immediately, so re-tune it.
By doing that several times, you're final tuning will stay longer.
As far as the fear of the string being tuned too high, this should never be an issue unless you are tuning the string significantly HIGHER then it should be.
But it still is a reasonable precaution to tune with the guitar "pointed" away from your face. :-)
Now back to the drop D tuning.
You didn't mention whether this was an acoustic or an electric.
For various reasons, including the typically lighter gauge strings, any electric will tend to be more sensitive then an acoustic.
But either way there are so many factors involved that affect tuning stability.
Many of them have to do with the quality and setup (setup meaning truss rod adjustment, bridge height adjustment, or a poorly grooved nut ) and all these things can collectively make the situation worse.
But honestly, if dropping the LOW E to D really made all the other strings go as sharp as you described, this instrument may have some serious flaws, and its going to be an ongoing frustration.
Don't let it discourage you...
only the luckiest individuals have a near flawless guitar when they start.
But working through it will soon make you an expert on all the little nuances of guitar tuning, and eventually you'll at least be able to get the most out of the instrument, despite its limitations.
Then when that's not good enough, it will be time for another one, and you won't need to rely on the shop owner to help you pick it out! :-)
For now though, even though many of us can help you here, its best to show the guitar to a more experienced player AND demonstrate the problem in person.
Without actually checking this guitar and seeing the issue in real time, its tough to know how much of the problem is just your inexperience, and how much is the fault of the guitar.
It is a frustrating (some even say wretched) instrument, but nothing beats the sound of a guitar. :-)
music.stackexchange.com/users/14830/Randy
I went to a guitar shop yesterday to buy a new set of strings and had them replace my old ones. The guitar sounds fine like the usual eadgbe sound but when I tried tuning it to drop D using a guitar tuner, I noticed the tune of the strings were like C#, D# etc. (I forgot the others.) I tuned it to EADGBE and it doesnt sound right. Any alternative ways to get it back in tune?
Difficulty tuning in Drop D
music.stackexchange.com
<guitar><tuning><strings>
music.stackexchange.com/questions/24173
music.stackexchange.com/users/13524/user3736846
The key to the issue of tuning problems with guitars is to learn to change and tune your own strings.  It is difficult to provide a precise answer without more detailed information. However, you would be best advised to get into the habit of learning to tune your guitar yourself.
Ideally, a guitarist should get into the habit of changing their own strings and learning to tune them themselves. It is difficult to provide a precise answer without more detailed information.