Please go through the following links to know more about moon and remedies to strengthen it:
Also, the following post contains the various problems what a person can face if the Moon lacks in strength:
Moon primarily is a sattvik planet. The remedies that do not involve spiritual bent actually will just strengthen your moon. Actually Moon needs to be with your Jupiter. The best combination that you can get in a chart is that of Moon in first own house with Jupiter (exalted). It would be better if the Moon is a full Moon or it is at 90 degrees with respect to Sun. Such a combination is capable of warding of any evil that would come on your way.
So, if you are serious to strengthen your Moon, please also strengthen your Jupiter. So in that case the best remedy will be to read the Geeta. It makes one's mind passive, fast. Also, avoid indulging in any sort of politics and be righteous. Be social, be honest, and believe everyone (I know it's hard, I still am trying to be like that). Accept your fault. Say sorry. Control your speech. Don't be arrogant. All these qualities will actually make you a humble person, yet extremely successful in the long run. The most successful people in fact are most humble.
I am enlisting the remedies of Moon again as below:
2) Wear a Silver chain in your neck with spherical silver pendant.
3) Use good amount of water for bathing.
4) Wear whites on Mondays.
5) Offer Saptadhanya to birds.
6) Try out Chandra Tratak - continuous gazing at moon. If possible chant 'Om Namah Shivay' or any Chandra mantra.
7) Read the ShrimadBhagwadGita.
8) Worship Lord Shiva. Offer water to Lord Shiva on Mondays. Chant the Shiv
9) Except for those who have exalted Moon (in Taurus), you can lend milk to small kids on Mondays
10) Have lots of juices. Give water to citrus plants.
11) Do not waste water.
Also for strengthening Jupiter, do the following:
1. Find a Guru or make someone a Guru in your life and seek His guidance. That person must be someone who can guide you spiritually. Be in His presence, and serve Him. The more the blessings of a Guru you have , more is the probability is that your Ju is getting balanced.
2. If by any chance, you do not find anyone, make Lord Krishna as your guide. It was He who guided Arjun through the battlefield of Kurukshetra when he was confused in delivering his duties. Now this actually means to follow the Bhagwad Geeta, read it daily and try to assimilate its teachings in your daily life. Lord Shiva is also considered Guru. He is the master of the Spiritual world,which lies just above the Material world, in which we mortals exist.
3. Read literature, healthy literature. It is best if it is spiritual text such as ShriRamCharitManas, Bhagwad Geeta, Koran, Bible or Zend Avesta. It must be something that must help an individual grow.
4. Teach, be a guide yourself. Explain things to people. Teach kids. Distribute your knowledge to others. Indulge in healthy discussions with others, related to the community/society improvement and spirituality.
5. Avoid politics; don't indulge in politics of any sort. Don't conspire against anyone. Be away from any such elements who indulge in politics.
6. Be humble. Most humble people have best of Ju in their charts. Learn to say sorry, even if it is not your fault. Leave your ego and arrogance.
7. Respect elders and people of greater age. Touch their feet and take their blessings more often.
8. Wear a haldi (turmeric) and/or kesar (saffron) tilak on your forehead daily.
9. On Thursdays worship Tulsi plant. Put a spoon of milk in its roots.
10. Worship the milder form Lord Vishnu, viz... Lord Ram or Lord Krishna on Thursdays. Since Ju is signifies kids too. many people have problems related to children too because if weak Ju. The best they can do is worship the Lord in their child form. E.g. do tratak with the Lord when He is in His mother's lap. E.g. Lord Krishna in Maa Yashoda's lap. For girls, who want healthy and obedient children, this is the first step towards their motherhood. And they must look at the Lord in the manner they themselves are Yashoda Maa!!!
11. Do upwas on Ekadashi tithi (11th day on brighter half moon month).
12. Avoid having salt on Thursdays. Have grains only once a day. Prepare flour halwa, which must be sweet and have it with your first finger. Put Tulsi leaves in that halwa preparation and offer to the Lord before having it.
13. Visit temples or places of worship, more often than ever. Sit there and feel the energy and sanctity of that place.
14. Donate books (specially spiritual) to people. Develop your own library of books.
15. Donate yellow clothes to your Guru on Thursday. Donate yellow sweets to your Guru or any elderly person on Thursdays.
16. Feed cows with wheat flour, gram and jaggery (gurh) on Thursdays.
17. Donate land/property/money for the development of places of worship.
18. If your ascendant is governed by Venus (Taurus or Libra), avoid having excess Haldi (turmeric). Instead donate it. Also, avoid having bananas much, donate them too. Plant a banana tree and water it daily.