Buddhists actually believe in several Gods?
Not in the external sense but in the sense that these are aspects of mind. They're also used for visualizations and symbolism to harness the power of the imagination to influence "reality". Here's a very good explanation.
The Buddhist way of living appeals to me and I can see a lot of sense in it. I would like to engage with it but I don't want to feel as though I'm disrespecting it. I can't force myself to believe in a God, nor necessarily the idea of rebirth, (at least, not in the sense of a physical rebirth - I can totally grasp the aspect of a mental, emotional, spiritual awakening), but these concepts seem important, (or irrelevant, depending on where you look). So what do I do?
Do what you wish, nobody should be forcing you to believe something. I would just suggest that your meditative experiences and life wisdom arein tandem with the things you're learning. For example, there's no need to intellectually grasp the deepest philosophical tenets of Buddhism if you can't even relax the tension in your forehead.
but I don't know what I'm expected to believe and practice in order to achieve this
what is the best way to meditate, be that with a focus on religious elements or focus on the self?
Object meditation (like breath or mantra practice) is best to cultivate concentration and calm abiding. With these qualities we can have spaciousness of mind, enough to decide which thoughts to follow rather than the thoughts dragging us around. It's all about taming the mind to rest in the pure and natural state of being. Many people will call this God or the higher Self (Sri Ramana Maharshi for example) but in Buddhism those terms aren't used.