At Will Employment - In General
An "at will" employee in the U.S. can be fired at any time for any reason without any prior notice or warning. Outside a union shop or a civil service employment situation, even an illegal reason for firing does not give you the right to be reinstated in your job - instead it gives you a right to sue for money damages.
Unemployment Benefits and Employment References
If you are fired for good cause (or you quit in a situation that is not a constructive termination), you are not entitled to unemployment benefits.
If you are fired without cause particular to your conduct (e.g. you are laid off in the employer's reduction in the size of the employer's labor force), or your are fired for a reason that is not good cause (e.g. you are fire because the boss is annoyed because you are such a goody two shoes that you always show up to work on time), you may be entitled to unemployment benefits based upon your term of service and earnings (short time employees are often not entitled to unemployment benefits no matter what).
If you apply for unemployment benefits because you assert that you were fired without good cause, and the employer believes you were fired for good cause, the employer can dispute that finding in a summary administrative hearing. Employers fight awards of unemployment benefits for employees who are fired rather than laid off, because it affects the employer's unemployment insurance rates (and because they care, for non-economic reasons, if the integrity of their stated reasons for firing someone are not believed, or if their reasons for firing someone are not considered to be legitimate grounds for termination by a government agency).
Even if you are entitled to unemployment benefits, these benefits are much smaller than your regular pay, and generally last less long than the period for which you were employed. A formula or calculator should appear on the Texas Workforce Commission Website.
I assume when you say that "They put a few fake tags on me" that this means that they stated reasons that would be valid "for cause" reasons for termination and you dispute those reasons apply to you, but please clarify if I am mistaken. If that is the case, it is likely that you would have to fight for any unemployment benefits you are otherwise entitled to in an administrative hearing as the company is likely to contest your claim that you were not fired for good cause.
This also means that if you seek new employment that they will give you a bad reference to someone who inquires about your employment (although many HR departments are afraid to do that for fear of defamation liability and will only confirm the dates of your employment and your position).
Wrongful Termination Lawsuits
Separate and apart from unemployment benefits, if you are fired, not just without cause, but for an illegal reasons (e.g. race, sex, and select statutory prohibited reasons), you may bring a wrongful termination lawsuit. Some of those reasons (firings related to discrimination by a private sector employer) require you to file an EEOC complaint and have it investigated by them first, other of those reasons (mostly whistle blowing statutes and breaches of written employment contracts that don't allow for termination of employment without cause) allow you to immediately bring suit for wrongful termination.
The legal status of firing someone because you complained about another employee's ethical violation depends upon the exact nature of the ethical violation. For example, a U.S. Supreme Court case decided this month (i.e. February/March 2018) held that whistle blower protections under U.S. securities laws apply to people who report securities fraud to the SEC, but not to people who report securities fraud to a supervisor in the company.
"My fellow employee was a lying asshole who acted unprofessionally (in non-technical sense of the word), and I complained about this conduct to my supervisor and the employer didn't care" standing alone, would not normally constitute conduct that is covered by a whistle blowing statute that could allow you to bring a lawsuit for wrongful termination of employment, although it might constitute a constructive termination for bad cause by an employer (if you quit) or termination for bad cause by an employer (if you were fired) for unemployment insurance purposes.
The legal theory behind the amount of damages that can be awarded in a wrongful termination lawsuit is a bit obscure. As a rule of thumb, six months wages is a pretty typical settlement amount in a wrongful termination case prior to a determination by a court of employer liability. At trial, there is wide variation in what juries award in wrongful termination lawsuits involving similar facts. Sometimes the award is minimal even when the jury finds that the employer wrongfully fired you, and sometimes the award is very substantial, amounting to many years of lost income in amount.
Contractual Payment Obligations
Generally speaking, unless a written contract provides otherwise, you have no obligation to return a hiring bonus and the employer has a contractual duty (and probably a statutory one as well) to pay you through the date of termination without deduction for a hiring bonus paid.
This includes any amounts, including bonuses, that the employer was obligated to pay you, although proving an entitlement to a bonus can be difficult unless that standard for receiving one is clearly defined and you clearly met those standards as a factual matter. Sometimes, you can even win a breach of contract award for a bonus that was not yet fully earned if the only reason that the bonus was not awarded was the employer's bad faith conduct.
You could sue to collect unpaid wages in a Justice Court (the limited jurisdiction court for small claims in Texas) if necessary, if the amount you are claiming is under $10,000. If you have a claim for unpaid wages in a larger amount or also have other damages, you would need to bring suit in the appropriate county court or district court, depending upon the amount claimed.