The Tiers

As you play the challenge, you'll notice that some restrictions seem more important than others. Indeed, a lucky roll for your founder's career can make the difference between a difficulty nail biting start and an easy cruise (though it is possible to complete the challenge even with the worst possible start - do not despair!).

Here's my list of "the best and the worst":

Tier 1 restrictions, if you get them early, almost guarantee a great start to the game.
You can choose your traits, which will often halve the time needed to lift a restriction (Ambitious and Workaholic for all careers, and others depending on what lifts you have). The other benefits (teens going to school, ability to teach skills, skill classes) though much less important are also very useful.
Master Thief
Effectively, once you lift Master Thief you can choose the order of your careers: check the paper, if it isn't what you want, leave it and try again tomorrow. Furthermore you can take time off without punishment - especially attractive if playing with Generations.
Military allows you to wander the town. It is a prerequisite for more restrictions than anything else. Both of those are important, but trumping that is the fact that once you lift Military, you can move out the Sims you no longer need, which means you never have to deal with uncontrollable elders or ghosts.
Allows you to lock wishes, which in turn allows you to buy a lot more lifetime rewards.

Tier 2 restrictions are very useful, but not as powerful as tier 1.
Allows you to sell things (very minor - you shouldn't need the money, and you can always dispose of objects by putting them in the family inventory black hole) and removes the restrictions on how many things you can build or buy per day (which is the main reason it earns its tier 2 spot). Also useful to lift as it is often a prerequisite (for example, for travelling to vacation destinations). Barely tier 2 really for the base game though.
Even in an Apocalypse your Sims need to make friends. Lifting Espionage makes that a lot easier, as you can now go and visit your neighbours.
Allows your elders to keep on contributing. Close to tier 1 for Free Will Off players, and definitely tier 1 for any prestige occult with a shortened lifespan.
The ability to use the telephone whenever you like is similar to Espionage in the way it enables friendships. As a minor benefit, it allows you to do the Chess Legend LTW much more easily.
Grants you considerably greater flexibility in choosing careers. Not as good as Master Thief because it's a lot harder to lift what with the requirement to do interviews (it's hard to meet this when you don't have an easy way to meet people). If you have already lifted Espionage or otherwise have access to a large group of acquaintances, Journalism becomes tier 1.
No more stinky Sims. But stinkiness is really only an annoyance; it's nice not to have it, but not worth giving up a tier 1 for.
Ghosts are very annoying. But if you lift Military early and therefore don't have any (or many) ghosts, this drops to tier 3 (the ability to collect life fruit and death flowers is handy, but not good enough on its own). For some occults, this may be close to tier 1 (or at least a very strong tier 2), even with Military lifted.
All the fun of electronics. You can of course raise Fun without Science, so it is only a tier 2. Becomes more useful the more you've unlocked, especially with Alien Technology.

Tier 3 restrictions are largely uninspiring. They are not bad, just not very good.
You can buy expensive objects. Not useless, just not very impressive compared to tier 2 options. It is a strong tier 3, perhaps inferior only to Communications.
You can now move large objects and train faster.This is merely a minor convenience; the main thing this gives you is the ability to score bonus points via Adorable Adonis.
So you can buy a better bed. Protection money is trivialized early in the challenge, though it is a bit annoying to have to keep paying it.
You can have large families without risking starvation or tedious micromanagement. Large families are useful because they allow you to complete the challenge in fewer generations. However, use of stashed (spoiled) food makes this avoidable, especially since with Education you can ensure your Sims can be Slobs and get to lick plates.
Parties are never required. It's nice for your toddlers to be able to gain a few more skills, but mainly this is the "change my outfit" restriction.
This is a good lift, but you cannot even attempt this without considerable preparation, so it's never going to be a high early priority simply because it cannot be. In most challenges this will be a mid- or even end- game lift.
You can build wherever you like. Not overly useful without also lifting Business, but the fact that it's part of the requirements to go on World Adventures (where you can effectively earn money or learn skills without consuming 'real time') is enough to make it tier 3.

The above table is, of course, merely my opinion. Especially in the lower tiers you may have your own ideas that are different to mine (though I suspect everyone would agree with Military and Master Thief).

As a guide, all else being equal pick the better tier whenever you have a choice. You often don't get a choice, of course. And if your founder marries someone at level 9 of Rock Star, nobody will think any less of you if you decide to clear Entertainment early.