Native English speaker (who's done only a bit of reading on and has an amateur interest in articulatory phonetics).
The easiest way to explain/demonstrate aspirated consonants to someone is this:
Note the puff of air you feel on your fingers. It's pretty strong and noticeable, right?
That puff is an "aspiration" in this context so this is an aspirated consonant.
Now, same hand position: * Say "spin" - you're still going to feel a little puff because that's the nature of the consonant but it should be significantly less.
That's an unaspirated "p".
The latter one is how the "п" in "спасибо" should be pronounced.
(Using Latin letters for now)
For the aspirated "T" sounds (and I think this probably applies to "D" sounds and across the board but at least possibly also with "N"):
English seems to use what's called the alveolar ridge quite a bit. To find it, place a fairly relaxed tongue against your upper front teeth and then slowly pull it back. The tip of your tongue will drag along a firm but fleshy/gummy shelf (I did say I was an amateur) until you reach a bevel and then a ridged cliff where it drops off (up?) into your palate. It will also become less relaxed in the process.
Right around that bevel is where many English letters like T, D, and N are pronounced.
Russian seems to be more dental. What the means is, when Russians pronounce these consonants, their tongues seem to be more at the starting position I mentioned above and almost never at the bevel (and less rigid at the tip). It's the position at the bevel (and the firmness of the tip of the tongue) that causes the aspiration.
Unfortunately [forvo](forvo.com) only has one pronunciation for University in your case, but you can still hear it there. But listening to the (much more numerous) pronunciations of тот and до свидания might help. It's the same mechanic at work.
I'm less sure of how to describe the К/Х sounds beyond "expand your lower throat and pronounce them deeper there than you might normally". But that's probably not overly helpful.