There's tons of ways to remove weeds from a garden. In most cases, the best policy is to just water the surrounding area of the weed (or wait until there's been a lot of rain - not an issue here in Seattle), and yank the weed out. The most important part is getting the whole root, or else the weed will grow back. Waiting for heavy rain is so the soil will soften up, making it so you can pluck the whole root as opposed to snapping it off.
There are lots of home remedies out there for "natural" weed killers. The ingredients in many of these include vinegar, dish soap, salt, etc. These ingredients are typically cheap and readily available. However, don't think that because its "natural" it won't have any negative effects on your garden. Be careful when using these chemicals on your soil. These chemicals have the ability to destroy not only the plant, but the soil as well, killing all (good) bacteria to devastating effect. For example, salt has the ability to neutralize your garden soil for up to 2 years. For a garden scenario, you'd just have to replace the soil. These ingredients typically are non-selective in plants/soil they kill off. Gardens are so close, that you may also have a lot of collateral damage by the chemicals getting in to other roots. So if you plan on gardening more (I assume so), don't use these chemicals in your garden. Its best to pull weeds manually.
Better scenarios for chemical-based weed killers are gravel driveway, cracks in the cement, or even fence lines. These are places you're positive you won't want to grow anything at and would prefer the area to be neutralized of plant growth and soil nutrition.
Overall, the best way is waiting for rain and manually pulling the weeds. Garden soil is high value soil, so the risk by exposing dawn dish soap to your growing soil is too great. Home made weed killers are much better suited for areas you don't consider high-value soil destined for future planting.