I have already made a Tamer which has quite a few costume pieces, since as a new player that started a week ago and only level 15 until i heard about the next season coming in the next few days (27th) so i was told its best to wait for that and after reading about it rewards seem pretty good

Currently only have time for one class at the moment and i haven't experienced the story or the game properly yet other than watching videos, guides and testing all classes in the trial area

because i already made a Tamer with costumes pieces i heard you can't swap these over if i made a new tamer for the season?

so would either have to make another tamer just to swap the season server or use another class and then timepiece my existing Tamer so i can experience the game normally + seasonal at the same time

Can i make a class, unlock the timepiece quest and use it on my Tamer?

i know its more beneficial to use another class and get to level 61-62 and then timepiece and you would have two high level characters, but im currently only trying to concentrate on one character so that i can get used to the game, mechanics, different uses of items, gear, story basically everything and i would like do to this on my Tamer

I would make another class and play it until 61, then timepiece to the tamer. See the other class as a "mule" in order to bring your tamer up to speed with exp, skills and gear. Usually we get some special rewards per season to level some char, so it will be helpful for you to just make that char in order to get all the rewards. Your tamer may need to sit and wait for a bit, until you can play it, but from a "future" perspective, making a new char would be ideal. If you really dont like that other char, and miss your tamer, you can still use the timepiece whenever you feel like it. 

If by "costume pieces" you mean the (mostly cosmetic) outfits that go in your pearl inventory, those can be worn by another tamer, in or out of season.  Nope - oops. Sorry. For that matter, the (functional) gear can be worn by another tamer too. So while you could do what Minarya said and timepiece over to your existing tamer later in the season, you could also make the first season character a tamer and swap almost everything to that anyway.  Then if you wanted to you could timepiece to your original tamer, and end up with two of them at the same level. Or you could view the original as abandoned and timepiece to another class.  Or none at all.

About the gear: there are limits to what you can use on a season character, until the character graduates from season. But by that time you may feel that the gear you built during season is better anyway, depending on what you already have.

I just double-checked and this is not correct. Once you equip a pearl item, it's character bound and you can't put it in storage to give it to another character.  There are ways to convert some of the functional outfits like Venia and I think Splat to a different class, but pearl outfits can only be placed in storage before they are equiped the first time.

Unless you know something I don't.

@Alnilam is correct.  Once a pearl costume is opened, it's bound to character and can't be placed into storage to move to another Tamer.

Since your Tamer is under lv25, you can use the Timepiece on another season character which will convert your Tamer to a Season character again.  However, you must level your Season character up to 56 before you can use the Timepiece.  If you want to experience leveling up from 15-56 on a Tamer, you'd have to create a new seasons Tamer.  I don't see anywhere that it says the Timepiece has to be used on characters of different classes, so you should be able to use Timepiece your new seasons Tamer to transfer everything to your old non-season Tamer.  I haven't verified it myself, but have no reason to believe it wouldn't work.

Wups, totally got that wrong, thanks for the correction. :(