If you opt for getting the shot avoid depo provera like the plague and opt instead for noristerat or levonelle - yes most people don't know there is more than one kind.
The nuva ring has good reports so long as you use it correctly - you may even remove it during sex if you wish - so long as you remember to replace it again afterwards.
Patch has poor reports, with a high failure rate.
As this is your first month on birth control - if I was your doctor i'd be suggesting you stick with it for the 3 months, because it can take a while for you cycle to settle down.
Yaz is relatively new on the market but has undergone extensive testing. It is quite a step up for oral contraceptives.
Mirena, which is one brand of the ius (intra uterine system) is a copper and/or plastic device which releases hormones inside the uterus.
It has a long life (some ius say 10 years - some ius say 5 years) but normal fertility is restored upon removal.
The ius is the development on the older iud (intra uterine device) which does not contain hormones.
The differences are that an iud often lead to heavier periods, while women on the ius often report very light or no periods.
Chopping and changing birth control before you give it a chance to settle is more likely to end up with heavy bleeding, break through bleeding and so forth.