It's totally normal to have irregular bleeding for the first 2 or 3 shots. Then you normally don't get any more periods whilst you're on Depo Provera.
I wouldn't advise taking other contraceptives too. Not hormonal ones, anyway. The reason you're bleeding is because your body is trying to get used to the new artificial hormone you're putting into it. It's quite a brutal & unnatural way, to start with. Pumping more fake hormones into it, might seem to help initially, but no one knows of the harm it may do to your body in the long term, because no studies have been done on taking the two at once.
If you get really constant & heavy bleeding, the doctor can give you something called Provera to try to stop the bleeding, but if you can sit it out & you're not constantly pouring with blood, I'd say to wait it out until it settles.
Being on Depo Provera alone is a huge does of artificial hormone. When you stop taking it, you may well find it takes up to 18 months to get your fertility back. It can also cause bone thinning & in later life, osteoporosis (brittle bones disease). It convinces your body it's pregnant, by giving a high dose of the hormone that convinces your body it's pregnant. If you then start throwing mixed hormone birth control at it, for example, they could work against each other & you could end up pregnant. Depo Provera needs your body to have more progesterone in it in order to work, & taking a combined pill as birth control (that contains both oestrogen & progestrogen) could end up causing an imbalance that leads to pregnancy.
Don't do anything without the advice of your doctor, for these reasons.
Are you aware of the need to take good quality calcium supplements, that contain magnesium & vitamin D in the correct quantities, & why it's so important to do that? This can prevent the bone thinning. Very important.