Before you all flame at me to search the forums, I have.
I found a general solution to the problem; valve timing.
But to my specific question, I couldn't find anything.

Recently I just changed my oil myself for the first time, not really a difficult job. However, after I changed it, I noticed a ticking noise occurs while accelerating (not sure about while in neutral, didn't think to check), at leas through the first 2 gears, and when I shut it off.
I've heard it in other cars before, I figured it was something to help it cool down or whatever.
It's not super loud, but it's just a quick ticking that lasts for a while after I shut it off, and I can barely hear it while driving.

I was wondering, the previous owner said he didn't use synthetic oil, so when I changed, I didn't use synthetic (someone said I'd have to mess with valve timing, and I didn't want to because I don't know how).
But now I'm wondering, did he use synthetic oil and the non-synthetic, is that causing this?

 

When the car has not been run for about 30 minutes, check whether the oil is low on the dipstick. Bring the level precisely up to the full mark. Does the ticking noise eventually go away when you drive the car for 10 or 15 minutes? If ticking only happens when the engine is cold, then the ticking is normal. You might want to try a different oil filter next time, as a good oil filter helps keep oil at the top of the engine when it is not running. I've read that Purolator filters are an excellent choice.

If the oil level is fine and you still hear valve ticking, then adjusting the valve clearance should help. Note that ticking/pinging can also be caused by bad gas. One gas station in my neighborhood sells gas that causes my engine to ping, so I avoid purchasing gas there.

 

The previous owner may have been using heavier oil to make the engine quiet. There is no problem using synthetic if you want to.

If it's valve noise, you will hear it at idle too. It comes from the top part of the engine. It will be worse when the engine is cold. If you have that, you should adjust the valve clearance (also called valve lash). This is a normal thing for Civic engines, once in a while the valves need to be adjusted to compensate for wear.

Ticking after you shut it off is not coming from the engine at all. It is the exhaust pipes or their heat shields. This is completely normal. If you have a ratting sound while driving, get under the car and find it, and secure those parts. Some people remove the heat shields entirely but this is really not necessary.

 

Ok, it's a Bosch filter, so I think the filter's alright.
I'll check the oil level again, definitely. Put in 3.5 quarts, no leaks, but after I changed the oil was when I started noticing it.
I'll check to see if I hear the ticking while idling tomorrow, but I'm pretty sure it doesn't.
And I thought about the exhaust, but the big thing for me was having just changed the oil so I figured I went from synthetic to non-synthetic, but wasn't sure if that's a problem.

And yes, it does quiet down after being turned off for a length of time, I'm not sure how long.
I'm not sure how to adjust the valve timing or clearance, I saw some diy's so I'll look through those again.

it's a 1995, so I think 5th gen

 

The clicking/pinging that you hear while accelerating may be detonation or "spark knock". For more details, click this link:

http://www.misterfixit.com/deton.htm

 

Ok, update.
I finally opened the hood and put my face into the bay.
I'm about 90% sure it's coming from the lower part of the engine, that general area.
I can't get underneath it though without stands.
And it persists after shutting off, so I don't think that could be spark knocking.
What inside the engine or in the engine bay keeps moving, even after the car is off for up to...~5-10 minutes

Oil level was okie dokie, as well.

I also didn't bother with valve timing/clearance, was hoping to find a definite solution (or an easier fix)

 

No metal parts inside the engine keep moving when it's shut off. The only thing that might be moving is bubbles in the coolant (because the cooling system is not properly filled), but that's more of a gurgling sound.

Really sounds like it's just your exhaust pipe or heat shields. Really nothing to worry about.

Ramps are an easier way to work under a Civic than jacking it up, unless the suspension has been lowered and the ramps won't fit.

 

No ramps work, that's what I've been using. It's just a pain in the ***, our garage isn't cleaned out and our street/driveway aren't level.
So it's a hassle.
But why would it all the sudden start after the oil change? That's where my concern is, I'm positive I didn't mess anything up.
But if you think it's exhaust pipe or heat shields, eh...that's alright
What would click with them though?