You need a new party objective.
A one shot adventure is designed to provide closure, and wrap up the majority of its loose ends, such that it can be enjoyed in its entirety in one session. If you would like to continue on from this adventure as a strating point, you're going to need two things.
1. A plot hook for the party to pursue.
This can come from a range of places. Use your experience with the party from the first game to entice them with a new goal. If you know the party is particularly greedy, have tales of riches in a far off land entice them. If they are altruistic, have a nobleman attending the tavern approach the new heroes and request their assistance in a nearby town. This new plot hook could be homebrewed, or you could use a plot hook provided by a published adventure, which will transition your game into the published content.
Another source for an objective is your player's backgrounds! 5e has a wonderful way of setting up possible future events by creating an interesting past. Do you have an adventurer who needs to avenge a loved one? Maybe you have an adventurer seeking to connect with their deity on a spiritual journey. The best part about this is that the players can decide which objective they want to pursue, and will be more motivated to see it through. This brings me to my other point however...
2. Your party has to want to continue as well
It's great that you want to continue on from a one-shot into something bigger, but you have to make sure your players are on board too. As you mentioned, one-shots lead to a "drive it like you stole it" playstyle, because the players know they don't have future games to suffer the long-term consequences of their in-game actions. As such, they may have created characters that they wanted to throw away when they were done, or made reckless decisions that the characters may not have otherwise made. Be sure to talk to your group about whether they wish to continue with the same characters and story, if they'd like to make some tweaks, or if they'd like to start fresh.
If you satisfy these two conditions, you can start building an adventure for the party to continue on with. If you are planning on moving into published material, be sure that the players are starting at the right level, and that they don't have any gear that might unbalance the module.