Since no single lens covers 20mm to 200mm with a wide aperture and stabilization, you might need to go for two lenses:
1. Tamron 20-40mm f/2.8 Di III VXD (Best budget wide zoom)
• ✅ Fast f/2.8 aperture
• ✅ Compact & lightweight
• ✅ Great for landscapes & low-light
• ❌ No image stabilization, but Sony IBIS helps
2. Tamron 50-400mm f/4.5-6.3 Di III VC VXD (Best budget telephoto zoom)
• ✅ Covers 50mm to 400mm (can crop for more reach)
• ✅ Has Vibration Compensation (VC)
• ✅ Sharp images and great autofocus
• ❌ No f/2.8 aperture (but excellent reach & stabilization)
These two lenses combined will almost cover your desired range.
Other Possible Choices
1. Sony 24-105mm f/4 G OSS (For a balance of range, stabilization, and quality)
• ✅ Optical SteadyShot (OSS)
• ✅ Sharp, versatile, and reliable
• ❌ f/4 is not as wide as f/2.8
• ❌ Doesn’t reach 200mm
2. Sigma 60-600mm f/4.5-6.3 DG DN OS Sports (If you want crazy reach)
• ✅ Optical Stabilization (OS)
• ✅ Super versatile zoom range
• ❌ Heavy and large