The best way to integrate your work into an art gallery is to be different.
Visit galleries near you and find out which ones have artwork that suits your style. Many galleries are run by a curator with a personal taste and you can take advantage of this. Research the chief curators and the work they present and ask yourself, "Does this person like my art? Think about your subjects, your approach, etc.
Show how your art is different from the work of other artists. This is difficult because your work may look like others, but it should not be too closely related. Curators are businessmen, so they won't put everything on horseback.
Go to openings (usually Tuesday nights) and networks. Sell your enthusiasm and talent. Get people in the gallery to want you and tell them that you are very interested in seeing your work there.
Go for promotions. If you apply for grants, you will expose yourself to other artists and curators on the jury. Even if you don't win, the art community sees that you're serious.
Avoid cost opportunities. Do not apply for exhibitions (or prizes) that require a submission fee. Such "competitions" are usually fundraising for the organization or gallery and offer little career opportunity or prestige (and may even make people take you less seriously if you include such an exhibition on your CV). There are occasional exceptions to this rule (especially in the case of Artist-in-Residence programs), but in most cases it is advisable not to pay people to view your work. Avoid paying galleries to attend an exhibition (or solo show). You will not find such practices in any reputable gallery.
Write an email request to the galleries where you would like to exhibit. Include as many examples of your work and information as possible. Also send a link to your artist's website. Include a brief statement from the artist describing your general attitude towards the artwork. Many curators like to collect documented information about an artist before showing their work.
Start an online gallery. You can invite other artists from your area or other artists with a similar style to yours.
Sign up for a collective artist gallery. Generally, these galleries are not limited to selling your work, but require a monthly subscription. However, it can be very difficult to become a member of such a gallery and you will need to submit your work and an artist statement to be included. Such collective galleries generally do not have a contractual arrangement that you must exhibit exclusively in that gallery.
Make a contract with a gallery. If you are accepted by a gallery, always insist on a contract. Galleries sell your work for you and take a share of it, so they are an agent, not a buyer. Make sure the share is clearly stated in the contract. It is usually quite high - 20-50%. However, galleries are also interested in selling your work at a high price, because the more you earn, the more they earn. Read all the contracts you sign, as you may have to exhibit and distribute your work exclusively through this gallery.