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The first step is to figure out what keyphrases you want your website to rank highly for. It doesn't make sense to ask for simply "higher rankings" because you can only get higher rankings for specific search keyphrases. This article will focus on Google, but the concepts are applicable to most search engines.
When you go to a search engine and perform a search, there are three places on the search results page that you see search results. Each of these places are areas where you can get improved rankings:

These apply only if your business has a physical location that you want people to visit. You'll need a free Google account. Start here.
You'll rank more highly if your business address contains search keywords. So if someone searches for "San Francisco shoes" and your business is called "JP Walters", and another business is called "San Francisco Shoe Store", their business will appear above yours. Other factors are also taken into account including: completeness of your Google profile, your proximity to the search location, recommendations and ratings, and links to your site from other related sites.
These are the traditional search results, which are the focus of most search engine optimization (SEO) strategies. You don't have to pay Google anything for them, but you either have to pay an SEO consultant some money to get you higher rankings here, or do a bunch of work yourself.
Getting high natural search results comes from two things: your site has lots of unique text content, plus other websites related to your industry (or other web pages within your own website) link to your site (ideally using search keyphrases as the link text instead of just "click here").
A blog is very useful in getting high natural rankings for a wide range of keyphrases. A blog is like a repository for search keyphrases -- you put keyphrases that you want to get high rankings for into each blog post, perhaps one keyphrase per post, and the blog does the rest.
In order to get high rankings for any particular search phrase, it has to appear in your website at least once, ideally more. The downside to a blog is that you have to commit to keep it updated, adding a new post (2 to 6 paragraphs) once a week or so, at least until you get the rankings you want.
As far as getting links from other websites, you should ask the owners of websites related to your industry to link to you. Some will trade links, some will sell you links. If at all possible, try to get a link that's not coming from their Links Page. Ideally, the link from their site to yours will be in their home page.
An alternative here is to get your site submitted to one or more free directories. While not as helpful as a home page link from a "real" website, they can help, especially if your site doesn't have much competition to begin with. You can submit your site to free directories yourself here, or I can do it for you for 25 cents each with a 100 directory minimum.
More grisly details on getting high rankings here.
For a pay-per-click (PPC) ad to appear in the sidebar of the search results, you have to create a Google account and set up an Adwords campaign.
While it doesn't cost you anything to have the ad show up, you have to pay Google every time someone clicks on your ad (from 5 cents to $50, but usually less than a dollar), whether or not they end up buying something from you. Money paid to Google can add up quickly if not done judiciously. Now you know why they're such a big company.
Details on setting up an effective PPC campaign here.
So basically, you can spend a bit more money and/or time up front to get high natural rankings, or for a quick boost you can start an Adwords campaign. Natural rankings fluctuate over time (usually trending downwards), so they do require a bit of upkeep from time to time.
I can help you set up an Adwords campaign, get you going with a blog, do SEO optimization, or submit your site to directories.