The key to growing plants is how much light they will receive. If you get a double fluorescent bulb for your 5 gallon tank, you should be able to grow medium to low-light requiring plants.
Anubias (dwarf varieties) Cryptocorynes (short varieties like 'wendtii') Valisneria spirilis, Java Fern and Java moss are all small low light plants. Stem plants like Hornwort, Anacharis and Wisteria will all grow, too, but may grow so fast you might need to keep only the growing tip and discard the older part of the plant.
In my experience, Amazon Swords are pretty tall growing plant that need lots of light. The one exception (or possibly just the only one I've succeeded with) is the Ozelot Sword which has leaves that are mottled red and green when young. It also stays rather short, for a sword plant, so it would fit in a five gallon tank. The other plants I've enjoyed working with are the "betta Bulbs" that are sold as tiny dried tuber in plastic packaging. These are members of the apogeneton family. Different varieties are included in the mix and some of these become very long-lived and beautiful plants.
If your local Petco has a good aquatic section. you should be able to find great plants there. Just make sure the plants you buy are truly aquatic plants. My Petco stores stock many plants that can take very wet conditions but fail if completely submerged. Liriope, Spathiphyllum, Dracaena sanderiana (lucky bamboo) and Pothos/Philodendron are some of these.
Remember that your petco will special order for you if there's a specific type of plant you want but don't find in stock.
(Photo above is my 30 gallon tank. Fish include Yo-Yo Loaches, Harlequin and Espei Rasboras and one Red Tail Black Shark. Plants include Hygrophila 'Temple' (on either end), Wisteria (ferny plant in the center left), Cryptocorynes (center and foreground) and Dwarf Anubias (foreground right). Lighting is double bulb fluorescent in a room with a southern exposure and lots of natural light .)
Best of Luck
Betta Fish with Peace Lily Plant: with live Betta fish
The first thing to ask yourself when you’re keeping betta fish in water-based houseplant environments is if you want to use live or fake plants. Both are fine, but you need to take some things into consideration.
10 gallon Betta Fish Tank with Potted Plants - YouTube
Beginners often start with fake plants for their betta fish and to make their tank more visually appealing. That’s perfectly fine even for those who don’t ever wish to care for live plants. It’s very, very important that you choose silk plants over plastic plants though unless the plastic is soft and not jagged. Betta fish have delicate fins which can be easily torn on hard plastic edges.
Are These Plants Betta-Safe? - Betta Fish and Betta Fish Care
Betta fish have a that allows them to breathe oxygen through atmospheric air as well as through the water. In their natural habitat, Bettas’ ability to breathe air allows them to swim in shallow water between many kinds of water plants, and rise to the surface to feed on larvae and plankton. Natural: One of the more common ornamental objects used in Betta fish tanks is natural water plants. There are many varieties from which to choose with some types more suitable for this species than others. If you decide to purchase natural plants, go to a reputable fish dealer who will know which are the best for your Betta.