It is a vine, and in warmer climates it can be quite invasive. Give it something to climb up and when it gets a bit long you can trail the runners back to the bottom and it will climb again giving a bushier effect. If you have it as a hanging plant you can just let it trail/hang down. Usually after hanging down a bit it will turn up and climb back up on itself.They've very easy to grow, just don't let the potting soil stay too soggy, or too cold.

mushibu10 - Ken refers to potting 'media' (probably ought to be potting 'medium' if you want to nit pick about singular and plural) but over here we call it potting 'compost'. Seedling compost isn't really the thing for long term growing as it doesn't have sufficient nutrition and is quite dense. I just use the stuff sold as house plant compost. Any garden centre or DIY place such as Homebase will have it. No need to repot now but when you do I'd get the appropriate compost. Note to US readers. Don't panic over me recommending 'compost'. In the UK the word means both rotted down stuff and growing media. It's quite clear which is which in the shops.

It needs to be repotted! As far as I know is that it can take a lot of your sun, needs good drainage and a fair amount of watering and likes humidity.

Purchased at a grocery store, my money would be on "Stargazer", a very popular Oriental lily. The Orienpet hybrid type "Anastasia" could be what it is as the flowers do look somewhat similar - but without any identification tag included with it, it could be whatever the supplier wanted it to be??? However, the short and sweet - if you bought it in bloom, each bloom usually lasts for under a week and the petals simply fall off leaving what is called a "candelabra" of spent blooms. This candelabra can be cut off. The lily only blooms once per season, oftentimes for a long time as each individual bud opens in succession - no amount of care will give you a second bloom, however. Don't think you did much with the self-pollination idea. Need an expert, but I think most lily cultivars are self sterile - they can't use their own pollen. Lilies are self contained units growing up from a bulb - the bulb pretty much has what it needs to grow, bloom, set seeds (if you let it), and the main stem to turn yellow/brown and woody. This dying back process is needed to replenish the bulb for next season. Watering heavily and often will pretty well insure the potted bulb will rot - they like well drained soil. Unlike here in cold climates where lilies can be left in the ground year 'round, in southern Florida lilies would need to be given a cold period called vernalization if you don't get long periods of sustained cold temps. Probably dug up and refrigerated for a period of 6 weeks or so. Lots of info on the internet by searching Orienpet lily Anastasia or just simply how to grow lilies in southern Florida.