If you really want to get into the Eucharistic theology thing, the late Fr. Hardon had some very good talks about the nitty gritty of the mystery. Or you can do a shortcut and read a lot of St. Thomas Aquinas, from the Summa Theologica. (Which is online and translated into English.)
But your head is going to hurt. There is just no way around it. Sometimes it will hurt in a good way, but…
However, if you want a shorter answer, you should be aware that “accidents” is an Aristotelian philosophical/theological technical term, which pairs with the other Aristotelian term “substance.” Neither of these things is equivalent to what you are saying.
Substance is the is-ness of a thing. What makes a dog a dog, and a particular dog, from the point of view of existence? It’s the substance of a dog.*
The DNA of a dog, and every outward characteristic of a dog from his bones out, are the accidents of a dog. So you can see that accidents are not anything like an illusion. They are all the characteristics and components of a particular dog, and they all happen to exist in the material world; but they do not constitute the is-ness of a dog. (Or at least, not from the standpoint of a philosopher or a theologian who is not a materialist.)
You can conceive of fictional situations like an alien shapeshifter that is perfectly disguised as a dog, right down to the DNA – because humans do think there is some difference between the accidents and the substance of a creature.
If a prince were changed into a frog, fairy tale style, the prince would have the accidents of a frog. He would hop, he would eat flies, his DNA would be frog DNA. But as fairy tales tell us, the substance of the frog is still that of a human prince. (Of course, fairy tales usually cheat and give the frog some of the accidents of a human being, like the ability to talk.)
So yes, the accidents of bread are real and continue to exist. If you poison the chalice or if you are an alcoholic, the accidents of wine can hurt you, because they are also real and continue to exist. But the substance of the Eucharist in either form is Jesus Christ Himself: Body, Blood, Soul, and Divinity. So what is there is not really bread or wine anymore, even though it has all the outward, material bread and wine characteristics. It is Him.
Transubstantiation is a lot weirder than fairy tale transformations or alien shapeshifters. That is why the more you learn about the theology, the more you will be filled with wonder (and your brain with headaches). But theology is worth studying, and it is fun too.
- I am not much good at giving good definitions of Aristotelian or Thomistic stuff, so you would be better looking this stuff up than listening to me.