I have been reading up on Paranoia a bit, though it's really hard to figure out what information I am allowed to have and what information I'm not. I want to get into a game, but I don't like the idea of belonging to a secret society or having any mutant powers. Is there an option to simply not be a mutant and/or not belong to any of the secret societies? And if I have to, are there any consequences to just not doing any of the secondary objectives my secret society wants me to, or not using my mutant powers?
Paranoia is a game where everyone is or appears to be secretly working against you, and you are painfully aware of this. (That's why it's called "Paranoia.") Secret societies and mutant powers are some of the main ways the game brings this about.
Mutant powers give you extremely useful abilities, but they're also things you could get declared a traitor for having if you get caught using them. Secret societies offer convenient advantages for members and/or threaten to kill you if you don't obey orders, but the missions they send you on are invariably things that could get you killed if you get caught. Both are things that you want to keep secret at all costs, yet which the other players could very easily find out about.
Of course, this means that you've got a strong motivation to find out the other players' secrets, so that you can use them against them. And they know you're coming, and you know they know you're coming, and that means the conflict is very likely to escalate until one or more of you gets killed.
Tl;dr: Be alert. Trust no-one. Keep your laser handy.
Anyway, to answer your question: There is no way to opt out of having a secret society or mutant power, because the game doesn't work as well without them. Barring GM intervention, there is no way to 'opt out' of the mess of intrigue that makes the game fun. (Several editions of Paranoia tell the GM to tell the players that not having a secret society or mutant power is a possibility. This declaration is an important part of the game and should not be skipped.)
The consequences for failing to fulfill your secret society's missions are up to your superiors in that secret society (which is to say, up to the GM). Sometimes this may mean a punishment of some kind; other times it may just mean you don't get the contraband equipment or useful information that would otherwise have been your reward. There may even be no consequences at all. Or at least, no consequences that they tell you about...
The consequences of not using your mutant powers are that you don't get the benefit of using those powers. Of course, you also don't risk getting caught using those powers, and don't risk those powers going horribly awry somehow.
Oh, and you mentioned that it's really hard to figure out what information you're allowed to have and which information you're not. That's true. Accidentally revealing you know too much is another source of hilarity in the game. A healthy attitude for an Alpha Complex citizen is therefore "I know nothing," sometimes even when someone is just asking for your name. And, on that note...
SECRET SOCIETIES DO NOT EXIST, CITIZEN. THE ALPHA COMPLEX'S CAREFUL GENETIC SCREENING MAKES MUTATION A PRACTICAL IMPOSSIBILITY. IT IS THEREFORE IMPOSSIBLE FOR YOU TO BE A MUTANT OR A MEMBER OF A SECRET SOCIETY. IMPOSSIBLE BEHAVIOUR IS TREASON. IF YOU BELIEVE SOMEONE AROUND YOU HAS DONE THE IMPOSSIBLE, PLEASE REPORT TO HPD&MC FOR MANDATORY MENTAL RECALIBRATION.
In the newest edition of Paranoia, it's possible to end up without a mutation or a secret society. Two of the fifteen secret society cards are No Secret Society. All the (non-blank) mutant power cards have a mutant power, but it's possible to trade away cards when bribing the player on your left during the Redefine step of group creation.
I wouldn't recommend a player give their mutant power, though, and personally I remove the No Secret Society cards when playing. The name of the game is Paranoia, and how can you be paranoid when you don't have any secrets?