I ended up having the problem of not being able to choose perks when I hit about level 10. Here's a list of the ones I recommend, by far the most useful ones IMO.
By far, one of the most useful perks is Educated (need Level 4, and Intelligence 4+). This perk means that when you level up, you get three more skill points to distribute between your skills. The earlier this perk is taken, overall the more useful it is.
Defense-wise, one the most useful perks is Cyborg (you need to be Level 14 with 60+ Science and Medicine). This perk adds 10% to your Radiation, Poison and Damage resistance, and also gives you 10 points to your Energy Weapons skill.
Size Matters (need Level 8, Endurance 5) is a fairly handy skill. Quite a few of perks give you 5 skill points each into two different skills, but Size Matters gives you a straight 15 points into your Big Guns skill, especially useful for the Flamer, Gatling Laser, and Fat Man.
If you're a freerunner at heart, Comprehension (need Level 4, Intelligence 4) is a good perk to take. With this perk, every time you read a skill book, you gain 2 skill points instead of just 1. It doesn't sound like much of a boost, but when you realise that there are 324 Skill Books in the game, that extra point per book really makes a huge difference.
Bloody Mess is also a very handy perk, and it's a fun one too. It's main tagline is that creatures you kill often explode into a "red, gut-ridden, eyeball-strewn paste". Alone, that sounds fun, but it also gives you 5% more damage with ALL weapons, so it's a useful skill. You need to be Level 6 for this perk.
Once you hit Level 16, the "Tag" perk is available. At the beginning of the game, in the GOAT exam, you end up "tagging" three skills, which boosts them all by 15 points. This perk allows you to choose a fourth skill (you can't choose one you chose at the beginning) for an immediate 15 point boost.
Strong Back is a perk that I've found useful, as well. It allows you to carry 50 more pounds of items, which can be very useful to a lot of players. You need to be Level 8, with Strength 5 and Endurance 5 for this perk.
There is another perk, Life Giver, which gives you an immediate boost of 30 hit points. This works out as quite a bit of damage, so it works out very handy.
Swift Learner is the last one I'm recommending. You only need to be Level 2 for this, and have Intelligence 4. It increases the amount of EXP you get by 10%, and it has three ranks -with all of them together you're getting nearly a third more experience from everything you do, from killing a creature to finishing a giant quest.
Hope that helps! :) For me I would go with the "Puppies" perk. When you activate it every time your dog dies she will have a full grown puppy waiting for you at your house.