Will I be able to grow plants with standard aquarium lighting?

I have a similar tank and all three types of plants do well, though a better bulb will certainly not hurt. Make certain and pick up a pair of midget suckermouth catfish - otocinclus affinis as they do a superb job with algae and will not "overcrowd" the tank. Just remember to work through the day/night cycle (usually I do 14/10) but you will notice how the plants are acting and should adjust accordingly. Hope it goes well