I would highly recommend learning QM (Quantum Mechanics). QM is science's strongest current theory for "how the universe works," and it has very interesting things to say about determinism. It will be very hard to debate determinism vs. indeterminism without catching up on several decades of QM.
(This is a ridiculously high level view of QM. I give this disclaimer because I am sure my natural English approach has technical inaccuracies. I recommend learning the math from a professional teacher to correct any misconceptions I may cause.)
QM models everything as a "waveform," which is nice and clean for individual particles, but gets messy quickly as the particles interact. It turns out that its very hard to measure the waveform; the mere act of measuring it changes its state. The fundamental limit of "classical" measurement is the uncertainty principle.
Consider this thought experiment: Pluck a string in a dark room. Take a flash picture of it, and develop that picture. You will see the string in some bent shape... whatever position it happened to be in when the flash went off. You cannot simultaneously measure its amplitude and its phase. It could be at a low amplitude, but at 90degrees, where the string is as outstretched as it will get. Or it could be a very high amplitude, but at a much lower phase angle, where the string has a velocity.
Now if we wanted to get more information, we could take a second picture, and do some math to determine how the string had to vibrate to satisfy both pictures. We could even take 4 or 5 pictures and get an even more confident answer, watching the string vibrate. We could eventually measure both the amplitude of the wave, and its phase. Then we could do all sorts of cool things. Radar is built on this principle.
At the quantum level, things get hairy. Consider photographing a string so tiny that the mere energy of the strobe disrupts its motion, like a gust of wind blowing out a candleflame. This is going to be harder. Each time we photograph the string, we disrupt it enough that the information in the photograph ceases to be very helpful. When we take multiple photos, we find each additional photo is not adding any more information! (As for why this happens, learn the real math. It's not just an empty claim; it is a well recognized effect of QM that frustrated many a deterministic scientist!)
What if we turned the strobe down? What if we made it weaker so that it doesn't knock the string around so much? What if we just had a quiet lamp generating light in the corner? Now we wouldn't disrupt the string, but we need a much larger exposure time to get some information. As a result, we can easily see the amplitude, but we lost track of the phase, as it blurred together.
This is the Heisenberg uncertainty principle there is a limit to how much you can know both the position and velocity of an object. The strobe version could give position, but not velocity. The bulb version could give velocity, but poor position information. There are an endless set of options in between, but none of them violate a fundamental limit. (This is not just abstract philosophy. These results have been observed many times, and are highly accredited).
So, back to our problem of determinism. QM does not actually defend nor refute determinism. However, it does have some harsh things to say about its limitations. These arguments are known as "interpretations," because none of them refute the other. They just look at the problem in different ways. We'll start with your least favorite, and move towards what I believe will be your favorite.
Indeterminism One perfectly valid way to interpret the results of decades of QM experiments is to argue that there are, at a fundamental level, some events which are completely indeterminate. At each of these events, the universe rolls a statistically perfect die, and determines the outcome. This is not a mere "coin toss," where the physics suggest an outcome. In this case, the physics literally suggests there is no known way to predict the outcome. Nothing in QM refutes this position. In fact, in my opinion, it happens to be the easiest position from which to start to understand QM.
Many Worlds Another perfectly valid way to interpret the results is to argue that, at each classical "event," two universes are created. In one, the event occurred. In another, the event did not. This brings in the determinism you are looking for, but at a cost: we cannot currently observe any of the nearby worlds that are on "almost identical" paths. There is simply no known physical or mathematical solution to "jump" between worlds.
Unmeasurability QM does allow a single universe with determinism, as you desire. However, it comes with a catch worthy of a genie in a bottle: you can't measure it. In theory, for every particle which has a value we can measure, we can think of it as being a particle with two values: an amplitude and a phase. If we were to be able to know both amplitude and phase, we could build your computer and start predicting the future. The catch is: we can't. By the rules predicted by countless myriad experiments, there is simply no way to classically measure both amplitude and phase of a value simultaneously. There are these cool things called "weak measurements" and "entanglements" which do really neat things to measure both amplitude and phase simultaneously, but even they cannot truly break free of this limit. Those cool structures like "entangled photons" have an Achilles heel: while you can be confident that both particles have the same amplitude and phase, observing it does dirty things to the link, preventing you from actually getting classical readings.
So there are three views on QM. QM does not refute determinism, but it doesn't prove it either. It does state that any deterministic view of the universe must hold to very strict rules, or it will be inconsistent with the observed scientific results of QM.
And to think: that was just discussing the idea of whether an electron has "freewill." Now imagine how much fun the discussion is for freewill of a person. Like with QM, it is totally possible for determinism and indeterminism to be consistent. All it takes is a few careful shifts of definition.
