In this guide I will recommend what I feel are low cost teams and refrain from listing any 6 star ++ banner characters that aren't highly recommended with the exception of Medea (I believe she's a good choice for second 6 star). I will cover teams for every element with a heavy focus on clearing engrave dungeons.
Wind
The first engrave dungeon is Wind. This dungeon should be easy if you choose Caesar as your first 6 star character.
- The first team is one you can build without any limited characters, but as mentioned above, you're likely going to need some levels on a dragon in order to clear f16 and/or the whole dungeon with this team. You will get free copies of Hua Tuo by completing the first 10 floors of this engrave dungeon, she is on the team to heal and take hits. You will get enough copies to 5, or possibly 6, star Da Vinci by completing your beginner missions and she is here to support your Octavius. You will get Octavius copies if you are a low spender, but you should wish list her to 6 star if you're not because she is the DPS for the team. Andersen is a free SR character that gives a nice attack buff at level 61 (it's fine if she dies). 2 copies of Beethoven can be obtained from the permanent chest mileage event (wait for the twice a month "Chest Opening Plan" event before opening chests). If you can't get Beethoven yet, use Chosukabe.
- For the second team, you're replacing 3 characters with the Caesar team. If you don't want to pull a copy of Kong Ming and Yoshiaki, it's fine to replace just 1 character (Beethoven) with Caesar.
Fire
The second dungeon to unlock is fire. This element has good budget friendly options for clearing all of the floors.
- The first team requires 2 ++ banner characters, but they can work at 2 or even 0 stars (ideally 2 though). Watt isn't important, but her passive allows her to survive fatal damage a few times which can be handy for survival. Michelangelo adds more burns to the enemy which can increase the success rate of this team and she works best with 4 stars, you will need to use the wish list if you want to star her up, but you can probably manage without 4 star if you have 2 star Prometheus and 2 star Takeda. Takeda is an essential part of the team (Honda works as a replacement, but don't summon for her unless you have her already) as she allows for the enemies to be stunned after you get 8(6 if she's 2 stars) burn marks on them. You will need to rely on ++ summons and/or the "Magical Encounter" event to get copies of Takeda. Prometheus can be obtained from the "Chest Opening Plan" twice per month event for 520 points and she will give 2 burn marks (4 at 2 stars) to all enemies and remove buffs from them. Merlin is your damage deal, but she doesn't need offensive stats or any investment at all, she just needs to survive (and not be disabled) until she uses her ultimate skill on turn 2. It's best if Takeda > Prometheus > Michelangelo (not required) are faster than your enemies.
- For the second team, you just use Zhou Yu over Merlin, but she requires damage stats and stars. Only use her if you summoned for her from her special banner as your first 6 star character.
Water
The third engrave dungeon is water. Water is a bit hard to clear without 6 stars or dragons.
- The first team requires at least 1 (2 are shown) ++ banner character and you should only use them if you happen to have them or if there is a banner with free pulls for them. Darwin isn't required (and shouldn't be used if Robin or Yagyu Jubei are giving you problems), but her summons can absorb damage and she gives a nice attack buff. You'll have to wish list this character and it's better if she has more stars. Romulus requires 4 stars to work best and she is also a character that you will have to get from your wish list to use. Catherine is the aforementioned limited character that is required for this team, she works fine for this purpose with even 1 copy and she can be a good tank for some bosses in the game, so it isn't a terrible idea to get that 1 copy when possible (you can also get her from the "Shining Star" summoning event). Akechi is your DPS option here and it's best if she has as many stars as possible because she relies on stats to do damage. Zhen Ji works fine with only 1 copy, she allows your DPS to use their ultimate skill twice (your DPS must have the highest attack out of your allies). For this team you want Catherine > Darwin > Zhen Ji > Akechi speed order, but Akechi doesn't need extra speed. You want Catherine to be faster than your enemies.
- For the second team, you need have either 6 star Medea or a levels on your water dragon. As Medea is a support, you should only be 6 starring her after you have your first ++ 6 star character. She is alone with Nene in order to guarantee that her buffs will only target Nene.
Earth
The final engrave dungeon is earth. This element can be easily cleared if you chose Medusa or Iori as your first ++ banner 6 star characters, but can be hard without them.
- The first team requires no ++ banner characters, but may not be able to win without dragon stats. Hercules can be obtained from the Hero Assembly shop and she can take hit and RNG sand cage enemies. Xu Chu is a wish list character that can stun 1 enemy, but it's not worth trying to get copies of her. Tamamo is your damage dealer here and can do a lot of damage with the earth dragon, exclusive equipment (EE) 50 and EE30 Hugo, but I don't recommend investing in her exclusive equipment because you won't use her much. Hugo is here to support Tamamo if you have invested in their EE for some reason and/or to stun enemies on turn 2 and it isn't worth trying to get copies of her. Himiko requires a certain amount of time to pass on your account before you can wishlist her. Himiko is here to support the team with sand traps (targets the enemy in the same position as herself) and do damage with her ultimate skill, it's better if she has more stars, but she can work well without any. It's good if Himiko > Hercules > Xu Chu > Hugo are faster than your enemies. I highly recommend waiting for one of the ++ banner DPS characters over this team, but if you're skipping them for some reason, you can use this.
- For the second team, you just add Iori. You want her to be EE50, 6 stars and have some speed investment ideally. This is who you use if you went for her as your first 6 star character, the other characters are only there to prevent enemies from disabling Iori and give her more balloon marks.
- For the third team, Medusa can be alone. You want her to be EE50 and 6 stars. Her EE50 allows her to counterattack, so she benefits from being alone. If you went for her as your first 6 star character, this is who you use.