Aquarium plants need three basic things to thrive: proper lighting, proper nutrition and many, if not most, require CO2 injection to really thrive (with some it's nearly essential).
First lighting, the so-called plant lights, gro-lights, etcetera, do not provide enough lumens for many plants, i.e. they don't give off enough light. And they don't always give off the proper wavelength for optimum growth either. The best thing I have found for plants is the standard 6500 Kelvin Daylight fluorescent bulbs available at any box store. For a 55 gallon planted tank I use 4 light fixture with T8 Daylight bulbs in a shop light type reflector. The wavelength and intensity of four of these bulbs will be more then enough for almost any plant in a 55, and too much for some low light species. Compact spiral fluorescent bulbs (same 6500K) can be used on smaller tanks or for addition lighting on long tanks in conjunction with the 4' fixtures; I have not found 6' T8 fixtures or bulbs available locally.
You should know that while this lighting will grow plants, it is not optimum for viewing of your fish; their colors look better under the reddish-blue light of most standard aquarium fixtures. But the look of a lush tank of thriving plants and healthy fish more than offsets this.
The next thing is nutrition. Many aquatic plants take nutrition directly from the water, some by their root system. You can enrich the standard sand or gravel by creating a "dirt tank" where there is a layer of organic soil under the gravel. Look this up and do research before you attempt this! With a dirt tank, you will get an algae bloom until your tank stabilizes that may last a few months so be warned. You can also use laterite gravel which, while a bit expensive, is loaded with iron which many aquatic plants need in abundance to really thrive. Plants love it! Then you should do some research and find a good liquid aquarium plant fertilizer and use according to directions.
All plants take in CO2 (carbon dioxide) and while it may be abundant in streams or lakes, that is usually not the case in aquariums. There are some plants that do not need additional CO2 like Sagittaria (I have them taking over a guppy tank and have to thin them regularly, they only get bright lighting, few nutrients and no CO2), but if you want to have a variety of species CO2 injection is a must. You can make your own with sugar and yeast (Google it) but it requires a lot of upkeep. If you have a good size tank it is best to buy a CO2 tank, and regulator with a diffusor and bubble counter. The tanks can be found at restaurant supply stores, the regulators and diffusors can be found on line, but they can be expensive. Look to spend at least $120 on tank, regulator and diffusor. The CO2 should be on a timer in conjunction with your lighting, you don't need to run it 24/7, only when the lights are on, plus it will save on refills. Refills usually mean exchanging the empty tank for a full one at a gas supplier, refills locally cost about $40, but can last nearly a year.
Once you have these three parameters in balance, your plants will not only grow, you will be thinning them out about ever month or two just to control them! The good news it you can take the extras to you LFS or give them to friends. I can't tell you how many I have even had to mulch or feed to the silver dollars!
Quarantine or sterilize you new plants before adding them to you tank, this will prevent certain types of algae from getting into your tank. There are several methods you can find on line, but because some can harm your plants, I will not recommend any specific one; do research!
You should also get an algae clean-up crew of fish for you tank: a few Bushy-nosed Ancistrus for brown algae, 3-5 Siamese (not Chinese) Algae Eaters which constantly groom you plants looking for algae and may help control the dreaded Black Beard Algae, and 6 or so Ruby Barbs for hair or thread algae. Even a few snails, but again do you research; some will devour plants. Let the clean-up crew do the work so you don't have to and so you don't need chemicals to control unwanted algae, plus Ruby Barbs are quite attractive and not belligerent like Tiger Barbs (which will eat plants!).
Good growing!