Bring a large pot of generously salted water to a rolling boil. Add pasta and cook until al dente, typically 8-10 minutes depending on the variety. Drain thoroughly and rinse under cold water to stop the cooking process.
While pasta cooks, prepare the vegetables. Cut cherry tomatoes in half, dice cucumber, chop bell pepper, thinly slice red onion, and halve the kalamata olives.
In a small bowl, whisk together olive oil, red wine vinegar, lemon juice, oregano, minced garlic, salt, and pepper until emulsified.
In a large serving bowl, combine the cooled pasta with all prepared vegetables. Pour the dressing over everything and gently toss until evenly coated.
Sprinkle with crumbled feta cheese and freshly chopped basil. For best results, allow to rest in the refrigerator for at least 30 minutes before serving to let the flavors meld together.