Let's say I had a hand held device that I could use to slow down or speed up time around me, what would the side effects be? I thought about it and my first thought was that all other particles would move slower in slowed time, so it would feel colder the slower time gets. The opposite would apply for speeding time up. This would make it impractical for me to alter time in an extreme way since I would probably die of extreme temperatures. What other side effects could this cause?
First off it depends exactly what your device does, does it influence your mind or your whole body? If it is your whole body then perhaps there is some variable sphere of influence. You could have a little bubble around you.
If it is just your mind then you would be limited by the speed on your physical reactions but your processing of information could be much quicker. You could walk into a room and have analysed exactly what was going on - think some sort of time assisted Sherlock.
If it is your body then your idea is right. Your body loses or gains heat from collisions with whatever is around you. If the surrounding material requires more energy to get up to the same temperature as you then it will cool you more (we see this when comparing being cold and wet to just cold - the water requires more energy to heat up, for other materials see this link on specific heat capacities).
It depends what your boundary is though, do your clothes feel the same time change or would they resist your movement? (Any material hit at a high enough velocity can be like a brick wall). If you include your clothes then you could design a suit to protect yourself - though I doubt you would be too subtle in it.
Standing still wouldn't cool you down when time is slowed - moving would though. Convective heat transfer only occurs between bodies in contact, when you move through slow moving air you will hit more air particles and having more in contact with you they would take more energy. Standing still you wouldn't experience this...unless things in contact with you experience your time you might heat up a little more because your body would be producing the same amount of heat but the particles of air would be hitting you less often and so you would lose less.
Other experiences: slower or faster aging (depending which experience you used more). Faster or slower information processing - this won't make you cleverer, only mean you can see what, given enough time, you would see anyway. Your reaction times would adjust - as already mentioned by @abc1234 - so don't stand anywhere busy when you speed time up or you might have something hit you (if your clothes are moving with you then the object hitting you would suddenly slow down too)
You may also look a little darker/further away - light hitting you when you're moving slowly will experience your time and be reflected back a little slower. This would be a relatively small effect though it depends on how much you slow time.
TLDR; Mainly it depends on where you draw your boundary: Mind, body or clothes and other things in contact with you. One thing you would always gain is faster processing.
I suggest some ideas from the book 'The Girl, The Gold Watch, and Everything' for the effects of slowing down time (some may also apply to speeding up time):
1) If you stay in one place too long you run out of oxygen; you have breathed all of it in that spot.
2) The light you see shifts frequencies; everything looks red during slowdowns and a blue shift happens during speedups.
3) Items that are normally feel soft such as water become concrete hard during slowdowns.
4) Accelerating/Moving other things becomes much harder during slowdowns, but keep their momentum. It takes a lot of strength to throw a baseball, but it is then moving at 500 mph in 'normal time' once you let go.