After you are read your Miranda rights (i.e., the right to remain silent, anything you do or say can be used against you in court, you have the right to consult an attorney, and if you can’t afford one, an attorney will be appointed for you, etc.), you can still be questioned. The questioning can occur without a lawyer present, but only if you voluntarily give up your rights.
You must also understand what rights you are giving up. For example, if your native language is Russian and you were read your rights in English and waived them, it could later be argued that anything you said couldn’t be used against you in court, because you actually didn’t understand your rights when you waived them.
However, if you are read your rights, and tell the authorities that you want an attorney, the questioning must stop. You must be direct and tell the police that you want an attorney. If you say something like “Maybe I should get an attorney?” that may not be enough to invoke your Miranda rights. If police are questioning you and you want it to stop, you need to directly tell them that you wish to speak with an attorney immediately.
Even though you can’t be questioned after you have asked for an attorney, you may be asked to take certain physical tests such as DNA or a test to measure the alcohol content in your blood. You should consult an attorney before agreeing to take those tests. In many cases, the test techniques are flawed, and they can create extra work for an attorney.