If your site has been submitted to Google, Yahoo and Bing, then that means it is on their list of sites to crawl for information. That could happen today, tomorrow, next week or next month. It just depends on each Search Engine’s schedule. Once that has been done, your site will start appearing in search results, but until then, there isn’t anything you can do to speed up your including in the search results.
What can you do to get your name out? Facebook or Google ads can work. You basically write them to target keywords of phrases that someone might type in to find a business like yours. When someone does use that phrase, your ad will appear. Normally you only have to pay for the ad if they click on your link, and you can set a daily budget so the costs remain under your control. (Once your daily budget it reached, the ad stops appearing so you don’t get charged any more for that day.)
Click for more information on Google Ads.
Also, I recommend setting up a Google My Business account. Enter in all your pertinent contact info, a description about your services and some photos, and then encourage clients to post reviews of your services on this account on Google. This creates the box on the right side of the screen when someone Googles your company. (If you go to Google and enter Digital Rose Design Studio as your search term, you will see what it looks like on the right.) My impression is that using Google My Business (reviews in particular) can help with your rankings on Google.