Welcome @teacher84, sorry you find yourself here. 
Different doctors have different advice on when to start trying again. Some say wait three cycles, some one. It’s down to personal choice really, but bear in mind how your are coping mentally as well as physically. I’ve had a chemical pregnancy immediately after miscarriage before, and it was very upsetting.
General advice is wait one cycle, but that’s more for ease of dating the pregnancy than anything else. Three cycle advice, will be again about giving yourself some time to grieve, as well as allowing your cycles to regulate.
Some say you are more fertile in the first three months after miscarriage, and therefore more likely to conceive again in this period. I would advise ensuring to you a clear negative pregnancy tests before trying, as the last thing you want is thinking you are pregnant to find out it is left over tissue from the miscarriage.
In terms of tests, you can ask your GP to run some basic bloods, check there is nothing obvious like an infection or thyroid issue. It will be down to your GP whether he/she will do this (some can be awkward). Mine did on my first miscarriage, but everything came back clear.
@Ells0204 is right, re more detailed tests. The general rule of thumb is three miscarriages before being referred to the Recurrant Miscarriage Clinic. Sometimes you may get a referral with two miscarriages if you are 39+ and/or have been trying to conceive for more than two years.
I’m currently awaiting a RMC referral after my third miscarriage last week. We have also opted for having the pregnancy sent away for testing. Hoping for some answers, but unfortunately sometimes there aren’t any.