Is a dirtbike a good beginner motorcycle, or should I get a cruiser? Any suggestions on bikes to purchase?

Get an enduro. They are a street legal dirt-bike. They have all the lights and a horn. They are the best to learn on. They have several nice models from several manufacturers that are around 250cc. Also, word to the wise. It's a good idea to get a lot of experience on rural roads or even off road before traveling the highways. I have had several bikes in my lifetime. All but one was an enduro. I ride mostly on rural roads and off road (way more than 100k miles). The only roadbike I ever owned I only put 10K highway miles on...and I had almost ALL of my close calls on that bike. People in cars just don't seem to see you. Highway riding is far more dangerous. The only reason I believe I had close calls and not serious accidents is because I obtained loads of experience off-road as a youngster before I ever drove on the highways. And a higher power was probably watching over me. Get some experience in lower risk situations and always be careful and anticipate what other riders and drivers may do. That's my advice.

If you wish to ride such a bike they are called dual-sports (street legal with limited dirt ability depending on make, model and year, some better than others). Take a MSF Basic Riders Course, learn from that and proceed.

if you are looking for something to ride on the highway, dirtbikes are illegal to drive cuz they lack the requirements to be a safe highway vehickle. (no headlights or signal lights, etc.)

Get an adventure tourer. This is a cross between a scrambler and road bile. Many manufacturers have such a model in their model range.

There are street legal dirt bikes, called Dual Sports. Suzuki, Honda, and Kawasaki make some of the best ones. The Kawasaki KLR250 or KLR 650, or the DRZ400 are good, cheap, workhorses that are extremely reliable and will easily fit into your budget used. They are easy to ride, but may be a bit tall for a shorter rider. A 125cc bike will NOT BE SAFE on the road... period. I have a dr125 that will MAYBE do 50mph, and takes a good while to get there. You want to be able to out maneuver the cage traffic, which includes acceleration and braking. If you plan on doing mostly street riding, then a Ninja 250 is a great choice for more of the sport style, and a Shadow 750 is a great choice for cruisers. Test ride a few bikes and choose the one that feels right to you.

The dirt bikes you are seeing around town are most likely "dual sport" style bikes. These look like dirt bikes but have the lights etc that make them street legal. They are great to learn on and have short wheel bases and tall suspensions...making them a blast to ride. They are tough, with plastic body work that is meant to get a bit scraped up. Not a bad thing for a beginner. You will ride it differently than a cruiser or sport bike but you will get a good foundation. You would need to take time to learn any new, larger bike anyway. And good for you for taking the riders' course, it is a great start...but just a start so keep learning while you ride.

The problem with asking what you should get in an open forum is you will get 500 different people answering and you will get 1000 different answers. My suggestion is get what you feel comfortable on. Don't make your mind up that a bigger bike will be harder to handle, in most cases it won't be. Just as if you were buying a car or truck, do some shopping around, ride several big bikes too, You'll find that bigger doesn't mean harder to handle, in most cases they are just as easy if not easier. The only recommendation I'd make is don't buy a crotch rocket until you have a couple years of experience. I'd pass on the dirt bike too, and opt for a smaller cruiser, say a 750..a honda shadow, or something in that size range..under 1000 cc until you are comfortable with a bigger bike. A dirt bike is fine for dirt but they aren't made for the streets or the interstates. at around 3000 to 4000 you can get a good used bike in that 500 to 750 cc range. You'll find that if you buy something smaller, you will outgrow it in no time and possibly lose interest.