What the best watering system for my vegetable garden?

A soaker hose and drip line system is the best and also the easiest. Attach a garden hose timer and you've got a great system. The hose kits are sold at the big box stores or online. This garden hose timer is on Amazon. Good luck.

I use these in my gardens with timers.

Andrews 50-Foot 2 Tube Sprinkler Hose 10-12348 https://www.amazon.com/dp/B00004R9UE/ref=cm_sw_r_cp_apa_js3YAbT681WHH

I would say soaker hoses would be the best system, but some people prefer the sprinklers which are faster. We save water from our roof to water our gardens by buckets also. Perhaps a combination of 2 of these methods would work best for you.

Depends on how sophisticated you want to go and how far away your garden is from your water source.

Soaker hoses are great especially with timers, you can really tailor your watering needs.

Sprinklers are a way to get the water out there fast for immediate relief of dry conditions, but they cover a wider area than you may want to apply water to.

A drip hose is the most efficient way to water. It has the added benefit of reducing the amount of evaporation.

Depends on your garden size, your weather type (humid or arid), etc. For us, we use a rain barrel hooked up to a soaker hose. It produces a very slow trickle. We turn it on early in the morning and leave it on for a couple of hours.