Kagetsuki

You can build an H-Bridge with any type of transistor, but the type of transistor will determine how the rest of the bridge should be built and what kind of input and load it can handle and how it will handle it. I'd bet most people build their first H-Bridge for motor control, and I'm guessing you are in the same boat. Just make sure you are connecting the correct collector to emitter combinations, make sure your diodes are facing the correct direction, etc. Also, make sure your diodes have a breakdown voltage higher than what you are using, and that your transistors are capable of handling that much load. Motors will wash current back when they are turned on, so diodes that react quickly will help protect the longevity of your H-Bridge. For small hobby motor H-Bridges, signal diodes and the like seem to work nicely. If even after checking all your connections and layout you still can't get it, try building something a bit simpler first to get your orientation down. An astable multivibrator is a great two transistor circuit which is quick to build and debug, and will get you used to the orientation of your transistors on a breadboard with essentially instant visual feedback if you plug LED's into it. If even then it doesn't work, double check your parts. Make sure each component is working properly by using a multimeter or building a test circuit. If they all work, make sure you aren't using heavy duty power diodes and little tiny transistors or some other combination that doesn't fit well. Also, there are a variety of single chip H-Bridges out there. They usually have heat issues and can't take a whole bunch of voltage or a big load, but they are simple and should work fine for smaller applications. I would definitely recommend heat syncs though.

The collector current being too low is usually easy to determine. If you have a lower load motor, put it on the line and try that. If you don't or that doesn't spin, try spinning the motor and see if it continues to spin. Most motors just need a higher current to start, and once they get going will take less current (assuming there isn't any high physical resistance like something attached to the motor). If it is in fact the case that there is not enough current then you need to either revise your design, incorporate some sort of charge-pump and use PWM, just straight up increase the power your power supply is putting out (like add another battery in series), etc. The thing about relays is they are electro-mechanical devices. That means that like your motor they have a variety of undesirable properties within an otherwise solid state circuit. There is however one very nice aspect to them though, and that's electrical isolation. The on-of switch portion of a relay is usually electrically isolated from the actual line you are switching on and off, so you can use a totally different power supply for your circuit on the other side. Then again, because relays are physical devices, they tend to have higher failure rates simply due to physical wear. Not to mention they cost quite a bit more and are often very large compared to other components. One solution to replace a physical relay is to use a sold state relay, which is actually a simple circuit that usually uses specially isolated light emitting and light detecting components. A quick and dirty DC version can be made with some tape, two LED's, a transistor, and some resistors. For applications like motor control when you need to control motors on a different power line and don't have abundant parts it's a lifesaving kludge. Though you may get laughed at, it works.

I live in Japan actually, not America. You can get the raw parts for a solid state relay for well under Y100 if you buy direct from a supplier, maybe just on or over Y100 if you want a triac version. Either way, if you have a transistor with a very low tolerance like you do making a solid state relay is probably not an option. You may want to note though that voltage divider circuits also divide current. Using 5 transistors stacked up would give you a theoretical 500mA max. Of course if the voltage is not divided correctly you could end up overloading one transistor, which would in turn unbalance the rest of the circuit and cause the other transistors to fail one by one. An array of 6 or 7 transistors with the best equalized voltage division you can get with resistors may work beautifully though.