What does a bearded dragon eat?

While dragons of the wild are likely insect and seed eaters primarily, they benefit from a varied diet of dark, leafy greens such as collards, kale, and mustard greens, etc. They also will eat hibiscus and dandelion flowers and leaves (all of which are good for them), and finely chopped or grated vegetables.
Even the leaves and flowers of geraniums are good for reptiles to eat. I have given these to green iguanas whom enjoyed these very much, but have to confess that my dragons didn't seemed too impressed with geranium leaves...

Do they eat insects and other bugs?

Bearded dragons readily accept crickets (age-appropriate for them. See data, below), and they can make up the basis of the captive dragon's diet, with mealworms, wax worms and king mealworms for variety for lizards older than one month. Be sure your feeder insects are themselves well fed. "Gut-loading" your crickets (allowing the crickets to eat afore-said dark green leaft items and veggies), will suppliment your dragon's overall diet. And it's a great way to induce 'finicky eaters' to eat their greens. If the coveted bugs taste like greens, -the dragon will be more likely to actually eat their vegetables when offered, sans the bugs... Crickets that are 'too large' to be safely ingested might cause an impaction, which is serious. Bearded dragons can and will swallow things that are bigger than their relatively small throats can handle. The large mass then impinges upon nerve endings in the roof of mouth and along the lining of the throat, and this can cause moderate seizures ("ataxia") whose neurological symptoms include thrashing about, gaping, rapid movements of body and limb, head-or-neck twisting as if staring at the sky (very common symptom), and 'shovel-nosing' their substrate. Usually, the syptoms go away on their own. Stress makes the symtpoms WORSE. So, avoid handling and cuddling the dragon if they manifest these symptoms. If the dragons will drink, this gives some relief. As might covering the cage with a towel or blanket. Probably the darkness calms the ailing lizard and reduces their immediate stress, and this probably is the biggest factor to relieve their suffering until the obstruction has passed or softened.

Do they eat meat?

Adult dragons may receive an occasional pinky mouse, especially beneficial for females after egg-laying. The 'quick protein' will help her regain her body mass and strength. Pinkie mice should be dusted with calcium suppliment prior to giving.
Because of bearded dragon's rapid growth, food items should be supplemented with a calcium/vitamin mixture (e.g., four parts calcium carbonate such as "Rep-Cal" brand calcium suppliment for reptiles) to one part reptile vitamin (Reptivite, Herptivite or Herpcare cricket dust), "dusting" food items "Shake-n-Bake' style. Do NOT give any other vitamins (e.g., for birds, cats, etc.) as it does not have to correct ratios of required vitamins and nutrients for your reptile and in the case of some vitamins, they have FAR too much vitamin-A for reptiles. Oddly enough, 'for people' vitamins such as "CENTRUM" are deemed to be more-or-less 'acceptible' for reptiles. Except for perhaps the vitamin-A content in 'for people' vitamins, one could in theory, use soley "CENTRUM" brand (or similar) for reptile vitamin/mineral supplimentation...

If dragons are kept on sand, food items should either be introduced in a smooth-sided bowl, deep enough to keep the food items in, but shallow enough for the dragons to climb in and out, or the dragons can be placed in a separate container (e.g., plastic sweater box) for feeding.

Feeding your dragon

The following is a suggested feeding schedule based upon approximate age:

Age: under one month:

Feed 2-3 times per day; crickets (small, dusted with calcium twice daily, every 2nd day with vitamin dust)

Age: One to four months:

Feed 2 times per day; (crickets approx. 2 week old, dusted once per day), occasional mealworm or waxworms, salad every other day.

Age: four months to 'mature' (7-12 months):

Feed 1-2x/day; crickets, mealworms, wax worms, pinky mice, King mealworms, salad every other day.

Age: Adult (over 12 months old):

Feed every 1-2 days; crickets (dust 2 feedings/week), Zophobas (giant mealworms), pinky to fuzzy mice (once per week or less often), veggies every other day. Avoid feeding "hairy mice" or "hoppers" as they are called. Newly weened mice (mice that may still be suckling their mother, but trying to nibble seeds and grains, etc.), are now like 'adult' mice, very apt to carry "pinworms". In the case of mice, pinworms are only passed from adult-to-offspring by ingestion. A 'hairy' mouse that is beginning to nibble seeds (and taste-testing eveything including fecal spoors of any conspecifics in the cage as well as accidentally walking through said spoors, etc.), will ingest, 'taste-test or at the very least, be 'exposed to live pinworms', increasing the risk of sickness in the future.

You can only be just so safe, but by avoiding this potential risk you might easily enjoy a very happy and healthy pet Australian Bearded Dragon for many years to come!