I normally shop at Target and was considering getting a Target Visa. Is this like a normal visa?

What they will check is to see if you have any upaid debts that are in arrears. They will check for employment or source of income. Do you have a job lined up for when you move to FL? Or are you planning on looking for one after you are settled there? If you are moving there without any employment lined up you may want to check into moving into a resident hotel. There is no lease and they are furnished. After living in one, you will have established a good track record of paying on time and have a good reference for when you do go apartment hunting.

Being so young and having no rental history the LL may ask for a co-signer. Or may not. It all depends on the landlord.

They may ask for two personal references. Maybe a bank reference to show you do not have a history of bounced checks or an overdrawn account.

Being from out of state when you apply for an apartment and having no rental history, will not work in your favor, but it may not necessarily work against you either.

I have no idea of what you mean by the term 'normal VISA'. The VISA corporation has many different types of programs to apply for. Here is a link to the VISA website to their student card page. Although Target is a very reputable company, you do not need to go through them to obtain a card unless you prefer to do so.

If you are still in school now you may be able to obtain credit card appications on campus. Even back in my day they had kiosks all over making credit card appications easily available to students. Right around graduation time the credit card companies send in reps to try to sign graduates up for cards.

I just rented a house to two recent graduates, both working full time in their field of study. One has lived elsewhere briefly, one has never rented anywhere. So between the two of them there was only one prior rental reference. One has a student loan and car loan; the other has only two credit accounts (a small clothing store and a visa), neither of which has much activity or any account balance to speak of. The two of them combined have less credit and rental history than most older adults I see. When I called the one prior reference I never received a return call. But I was glad to get them as tenants. Why? They handled themselves well. Dressed tidily, behaved in a mature manner, asked questions about rules, what I expected, asked "what if" scenarios, etc. That shows a willingness to respect both my requirements and the property. They both came across as quite responsible. I did not ask for a co-signer. Being a college grad is a plus. Not being up to their eyeballs in debt is another plus. Working full time in their chosen career field, huge plus. Having a roommate to share expenses, allowing more financial breathing room, allowed me to rest much easier (it's a large house so rent+utilities is higher than smaller houses or apartments).

On the flip side is a young relative who rented her first apartment last year in a large complex and she needed a co-signer even with a roommate (both about your age). First apartment for both, my relative has opted to work full time and pursue college part time. However, she made some stupid credit mistakes between 19-20 yrs old which she still has not addressed and that is going to haunt her until she gets it cleared up. She started out small (e.g. Abercrombie credit card). Then the credit card offers started rolling in and she took about everyone of them. She doesn't owe a huge sum but a little here, there and everywhere adds up and she's been very irresponsible about paying on time. I fear this is one lesson she is going to learn the hard way, because she fails to heed warnings of family members who care about her welfare (present and future). Someday soon she's going to have to stand on her own two feet without a co-signer and if she doesn't get her act together she's going to find doors closing in her face left and right.

So in answer to your original question ;) If you feel you are responsible enough with money, go for the Target Visa and request a low credit line (say $500 max, to keep you out of deep trouble). For the record, whenever a credit card raises your credit line you can immediately call them and request that it be lowered back to a limit comfortable for you. I have an "internet only" visa that I want kept at a low credit line because of a theft problem I had online. They have pushed it as high as $15,000 and I repeatedly ask them to lower it back down. (They'll send you a letter, "congratulations, you now qualify for a higher line of credit and we have increased your credit line to $xxxx.xx." Suzy Ormon would disagree, but I promptly call and request the credit limit be reduced to $500 again.)
If you feel you aren't sure how you'd handle a credit card, go for a department store or gasoline credit card, but don't take their Visa, just the card that can be used in the store or at the gas station. Bottom line: at your young age, don't let your credit spiral out of control or neglect to pay on time. For someone so young, getting started down the wrong path does not look good. You are the one who can best determine how well you handle money. Don't let yourself fall into the trap of "oh, I'll be sure to not go overboard". It's very easy if one does not exercise self control to get in over one's head, and fast. I've linked to a page below, you can scour for cards that offer low rates (personally I feel a low rate card is better than one that offers cash back incentives, because you'll save much more $ in the long run. And be sure to read the article about average credit card debt in the right side.

Congrats on your impending graduation! ;D

My very first apartment was one that I sublet from another student I met at the university, who was going to be away, out of the country, on an archeological dig for 8 months. The person who's apartment I was going to be living in trusted me with the place including all of their belonging. All I had to do was sign some sort of "occupancy" form with the management company so that they knew who I was. It was nothing more than my name as I remember. No credit check. Nothing. I was suppose to pay the leasee, and they were to forward the money to the LL, but I just sent the check directly to the LL and nothing was ever said about it. The person who's name was on the lease was still the 'responsible person' regarding everything in case something went wrong. Nothing did. It all worked out fine. I know it sounds like an odd situation for a first apartment but sometimes opportunities like that come up and it's good to send feelers out, and keep an ear pealed for something like that that may come up.