Focusing on GRE, subject GRE, TOEFLS, IELTS before applying?
For grad school, the GREs and TOEFL are treated more as thresholds. Once you cross the target score it may not have much of a bearing on your admission. For the general GRE, do some of the sample exams provided by ETS and see how far your score is from the prescribed targets. You can count on improving at about 5 points per week (so if you score 310 and your university needs 325, plan to study for three weeks). The TOEFL shouldn't be an issue for you based on your other Academia posts. The subject GRE requires at least 2 months of serious preparation. Again, try to see where you stand.

Try to do the exams at least a month before the application deadline, as that is how long it could take for the scores to reach their destination. If you are applying for a competitive program, you will need all your scores in place. Since it is only June, you have enough time to apply this year (to join next Fall).