I am working on an Orthodox conversion and starting to take on mitzvah's I have been looking at getting tzitzits and as i don't have a family background i see they come in different types such as Chabad, Sefardi, Ashkenaz and I've seen Avodas Yad. also whether I should be wearing wool or cotton. I am not sure which style is appropriate for me as a convert. Any guidance is appreciated.
Caveat, this answer is mostly what I've gathered from the communities I live in as well as discussion with my rabbi.
This all depends on the community you're converting into and the customs of your rabbi. If you choose to adopt Ashkenazi customs, you should wear tzitzit which are tied according to the Ashkenazi practice. The same formula goes for if you decide to become Sephardi or Chabad (most other Chassidim tie Ashkenaz as far as I've seen). In regards to the garment, wool is viewed as biblically mandated by all sources, although cotton has no such widespread appeal. I personally wear a wool garment with thick tzitzit tied according to the opinion of the Ba'alei Tosafot (techeilet).
Another point on vocabulary: "עבודת יד" (Avodat yad/avodas yad) means that the strings are completely handspun, they can be either thin, medium, or thick.
Another issue, although not one to get into immediately is whether your adoptive community accepts techeilet, as it is both an expensive proposition and a more complex issue when it comes to tying.
May HaShem bless you as you seek to come under the wings of His shekhinah!
In answer to your question(s):
According to halakhah, if one is not wearing tekheleth in his ssissiyyoth then all that is required is the correct amount of kasher strings and at least three windings (and there is a LOT of leeway in the halakhah for doing more windings, knots, etc.).
[See Rambam, Mishneh Torah, Hilkhoth Ssissith 1:9]
If you are using tekheleth then you need to make a decision as to which method you will abide by - and your decision need nothing to do with "custom" since neither Ashkenazim, Sefaradim, Hasidim, nor Yemenite Jews ever possessed tekheleth when their traditions developed.
There are about four methods that seek to be authentic to Hazal, the rest are relatively late and based on doubts regarding actual requirements, ignorance of how they should be tied according to the various beraytoth, combinations of differing (and sometimes opposing) halakhic views, and latter-day derivations and contrivances from the "kabbalah."
The four shitoth that I am referring to are:
[A] Ra'avadh (which is really the shitah of Rav Natronai Gaon)
[B] Rav Natronai Gaon (in his own words)
[C] Rav Amram Gaon (from his siddur)
[D] Rambam (which is a variation on the shitah of Rav Amram Gaon)
[See instructions on how to tie these on www.tekhelet.com]
However, me-iqqar ha-din (from the strict standpoint of the law) any of the extant methods of tying with tekheleth are kasher and one may make a blessing on them, but one should be aware of the relatively late and contrived nature of most of them.
All of this being said, any style of ssissiyyoth that personally suits you and accords with the halakhah will do.
The only practical note I can offer outside of halakhic considerations is that a great many batei din will refuse to convert - or will make the process much more difficult - for candidates who "choose for themselves" so it is best to just simply ask your sponsoring rabbi which one to wear. Doing such to prospective converts is petty and without legal basis - and is almost certainly a crime in the eyes of the halakhah - but the unfortunate fact is that things like this happen all the time. So, just direct these questions toward them and you should be fine. After you convert they no longer can dictate such things (read "micromanage") and you will be free to express your Jewish faith in the halakhic manner you so choose. Until such a time, my sincere advice is to play it safe.
As regards the material (e.g. cotton or wool), the ssissith should either be wool or from the same material as the garment itself (cf. MT, Hilkhoth Ssissith 1:1, 3:5-7). The only exception to this rule according to the halakhah is that no tekheleth should be placed on the ssissiyyoth of a linen garment, and this is by force of rabbinic decree (Ibid. 3:7).
Lastly, wear a talith qatan that is comfortable for you, regardless of the type or style of the ssissith that you end up attaching to it. Don't buy into the idea that a wool garment is the more religious or "frum" choice - it's not. If you want wool, then by all means buy a wool garment for daily use, but if you find it to be uncomfortable (as most do) then wear something made of cotton or mesh. Remember that while the fringes are certainly of spiritual significance, the garment mentioned in the Torah is of a practical one. We are not Mormons sporting magic underwear, we are Jews who attach a decorative fringe to the corners of our garments - as God commands us - which reminds us of our duties as part of the holy nation.
Hope this helps. Kol tuv.
which ever specific minhag you go by they should be wool as other ones are questionable whether they are considered a garment that require tzitzis. Avodas yad could be any minhag also, it justmeans that the strings were made by hand rather than by machine.
It's important to have a mashpia (mentor) that can help you decide your minhagim. Additionally minhagim are not just a specific way one does something within Judaism but are part of a specific pathway in serving G-d (a derech). It would be advisable to understand that and how you will be proceeding before completing your gerus.
I strongly recommend speaking to the Rabbi who is guiding you in the process before you get to the step of meeting with the bes din.
wishing you much hatzlacha