Because of how dramatically the Ministry changed after Voldemort took over, I'm going to split this answer up into two parts.
We don't have a ton of information about the wizarding legal system, probably because that's something a young Harry wouldn't have been very interested in. Because of that, it's hard to firmly say whether or not the actual laws discriminate on the basis of either wealth or blood-purity.
What's easier to argue for is a version of indirect discrimination, where the law isn't itself discriminatory, but is applied in such a way that the discrimination happens anyway.
Arthur Weasley seems to believe that Lucius Malfoy pays off Fudge for political benefit:
'What private business have [Lucius Malfoy and Fudge] got together, anyway?'
'Gold, I expect,' said Mr Weasley angrily. 'Malfoy's been giving generously to all sorts of things for years... gets him in with the right people... then he can ask favours... delay laws he doesn't want passed... oh, he's very well-connected, Lucius Malfoy.'
Order of the Phoenix Chapter 9: "The Woes of Mrs. Weasley"
There are a couple of things to note about this situation, though:
Arthur is very biased. Although it certainly sounds like a Malfoy to grease the political wheels for his own benefit, considering the animosity between the two families it's not quite reasonable to take Arthur at his word.
- On a related note, we do see Lucius Malfoy manipulating the interpretation of rules, but in all cases he coerces consent with threats of violence. For example, when he gets Dumbledore sacked in Chamber of Secrets (emphasis mine):
"Well, you see, Lucius," said Dumbledore, smiling serenely, "the other eleven governors contacted me today. It was something like being caught in a hailstorm of owls, to tell the truth. They'd heard that Arthur Weasleys daughter had been killed and wanted me back here at once. They seemed to think I was the best man for the job after all. Very strange tales they told me, too.... Several of them seemed to think that you had threatened to curse their families if they didn't agree to suspend me in the first place."
Chamber of Secrets Chapter 19: "Dobby's Reward"
We never see tangible political benefit come out of Malfoy's money. As I mention above, Lucius Malfoy tends to get his way through intimidation rather than money. If there are exceptions to that rule, we never see them.
What Arthur describes is basically political lobbying. Bearing in mind that Arthur is not an unbiased source of information, there are a lot of similarities between the two descriptions. Although "lobbying" has become something of a dirty word in politics, it's not necessarily unethical. The major issue comes about when there's nobody lobbying the other side, and we don't have a lot of information to suggest that this is true in the Harry Potter universe.
We do see that he gets A+++ seats at the Quidditch World Cup:
Fudge, who wasn't listening [To Lucius Malfoy disparaging the Weasleys], said, "Lucius has just given a very generous contribution to St. Mungo's Hospital for Magical Maladies and Injuries, Arthur. He's here as my guest."
Goblet of Fire Chapter 8: "The Quidditch World Cup"
And of course he also gets away with being a Death Eater, although the extent to which this was due to either his wealth or his blood-status is unknown.
There are tons of barriers for Muggleborns, it's virtually impossible to list them all, but a handful off the top of my head:
The exchange system is whacked out. Normal Muggle exchange rates are determined by the foreign exchange market; vastly oversimplifying, the GBP1 has a high exchange rate relative to the USD2 because more people want to own pounds than dollars3. One of the reasons this works is because the forex market is huge, so the ability for a single nation to affect exchange rates is negligible4. But in the Wizarding World, there's only one buyer of muggle currency: Gringott's. Gringott's has such enormous influence over the exchange rate, and the Muggles have a limited ability to balance that influence. It's a recipe for abuse.
School is a completely new environment. This one seems pretty obvious: wizard-raised children have had eleven years of exposure to the magical world through their family and friends, which gives them an incredible advantage in Hogwarts. Muggle-raised children, on the other hand, went to muggle primary schools and have muggle skills like arithmetic that are now utterly useless to them.
Ability to practice magic. Because of The Trace, underage witches and wizards are supposedly not allowed to practice magic outside of Hogwarts. The (notoriously unreliable) Harry Potter Wikia suggests that this is more true for muggleborns than for others:
The Ministry trusts magical parents to properly discipline their children if they perform magic due to the fact that the parents' own magic will constantly interfere with the Trace. Children who grow up in the Muggle world, such as Harry Potter, are more closely monitored
In fairness to the Wikia, I remember reading something about this as well. I can't find a quote at the moment, but commenters feel free to help me out.
Accessibility to Wizarding services. This is as much of a question as anything else. We know that someone from Hogwarts comes to muggleborn families to explain about magic and how to get to Diagon Alley and such. But what about the Ministry? Are they informed that they can get into the Ministry of Magic through a phone booth in Whitehall? Do they have to travel all the way to London for that, even if they live in, say, Ireland? What about getting into St. Mungo's? Getting to Platform 9 3/4? The Wizarding World is organized in a way that presumes instant (or at least inexpensive) transport between any two locations, which is dramatically less feasible for muggleborns, especially school-age muggleborns, and their families.
It should be obvious that none of this discrimination is actually codified into the laws, but rather that they're unintended consequences of a system that was designed to make it difficult for (wizard-hostile) muggles to penetrate magical society.
Once again, codified wealth discrimination is difficult to determine. I don't recall any particular examples of the Voldemort-controlled Ministry instituting laws that protected the wealthy.
However, the codification of blood-discrimination is undeniable. The Muggle-Born Registration Commission is exactly this, and doesn't pretend to be otherwise:
"Muggle-born Register!" [Hermione] read aloud. "'The Ministry of Magic is undertaking a survey of so-called "Muggle-borns" the better to understand how they came to possess magical secrets.
"'Recent research undertaken by the Department of Mysteries reveals that magic can only be passed from person to person when Wizards reproduce. Where no proven Wizarding ancestry exists, therefore, the so-called Muggle-born is likely to have obtained magical power by theft or force.
"'The Ministry is determined to root out such usurpers of magical power, and to this end has issued an invitation to every so-called Muggle-born to present themselves for interview by the newly appointed Muggle-born Registration Commission.'"
Deathly Hallows Chapter 11: "The Bribe"
While this may not appear to be a legal change at first glance, bear in mind that the commission is sending people to Azkaban for stealing magic:
The lift doors closed and as they clanked down another floor, Mr. Weasley said, "I hear you had information about Dirk Cresswell."
"So Dirk Cresswell is ten times the wizard you are," said Mr. Weasley quietly, as the lift sank ever lower. "And if he survives Azkaban, you'll have to answer to him, not to mention his wife, his sons, and his friends ."
Deathly Hallows Chapter 13: "The Muggle-Born Registration Commission"
1 Great British Pound
2 United States Dollar
3 For the record, this isn't completely true. This is more-or-less what happens in a floating exchange rate system, which is true for my example countries and for most other economies. However some countries, largely African economies, operate on a fixed exchange rate system, where the exchange rate is tied directly to something else, whether that's a commodity (commonly gold) or another nation's currency. In modern times, this is used mainly in unstable economies to keep the value of their currency relatively stable
4 In theory, anyway