What is the best way to lead a new soldier who is uncertain if they even want to be in the army?

PVT (Join to see) ,
As I am sure that my response will echo many of who have already posted. However, from my perspective with regards to your circumstance as this really is a challenging, yet sensitive issue for all leaders who posses empathy and want nothing less than the best for their Soldiers/team members. Before deciding, ensure that you do a full circle perspective before placing any type of judgement. First and foremost, if your Soldier has the definitive mindset that they are ETSing with no signs of compromise or negotiation, do not scoff at the decision. I say this because I have seen it far too many times during my 17 years where both peers and leaders will scoff at a Soldier's decision for wanting to exit military service coupled with the assumption that will not amount to anything in their next chapter; as if the military is the be all, end all. However, if the Soldier is uncertain on their decision, this is the opportunity to ground them back towards memory lane to the civilian on the street that walked into the Recruiting station looking for a change. What I mean is this: When most Soldiers join their respective military branch, it normally comes with a set of desires aimed towards personal goal achievement; albeit money for college, embark on their own independent career, provide for their family, and a wide array of other decisions. This is the time to gage the posture of that Soldier and their individual goals. This normally leads to open ended, fact finding questions such as "Why did you join the Army?" The Soldier will reply because of this reason...From there you ask, "Have you accomplished those personal goals that you set forth thus far?" Over 90% of the time, the answer is no. This serves as an opportunity for you to get them refocused on why they joined, and to use the next reenlistment opportunity as a chance to achieve those goals. After that option, then they can make a solid decision on continuing to serve or walk away. At least this will give them a peace of mind towards the next decision and ultimately prepare them for both their current career and the next chapter. Sorry for being long winded but I have seen this scenario play out too many times in which I both take to heart and have kept them to not only stay on the team and continue to provide their critical contribution factor, but to keep them honest towards their personal goals.
DISCLAIMER: THIS IS NOT A ONE SIZE FITS ALL BLANKET STRATEGY, AS SOLDIERS AND RESPONSES WILL VARY!!
Hope this helps and best of luck!

I think leading by example is the best way. Give them positive acknowledgement for an assigned task well done. Not everyone will be cut out for Army life but don't give up on a Soldier too soon.

Something I see too often which leads to soldiers wanting out is the army either not providing opportunities that should be available or administrative errors that derail career progression. I've seen a specialist miss the promotion list (who should have been an E5) by 2 day because S1 sat on the paperwork, bonuses that are years overdue, promised schools fall by the wayside due to lack of funding, ect. The army should be a very professional organization that provides soldiers every opportunity to achieve. If the organization is screwing with soldiers financially or in career progression, it's hard as a first line leader to change the perception that the army can't deliver (when honestly the soldier's frustration is warranted.) If the army wants to retain good soldiers and morale, they need to deliver on promises.