We have already used modal verbs several hundred times without explaining their use without any problem. Normally the meaning can be deduced with the help of didactical motivated translation. The system is the same in all roman languages, but the system differes a little bit from the system of other languages. Roman languages make some distinction not made by other languages and sometimes other languages make distinctions that are not made in roman languages.
Poder means that someone in a concrete situation is able to do something because there is no hindrance to do it. However there is a difference compared to English. "I can write" in English can mean two different things. Someone is able to write because he has learned it at school or that in a concrete situation he is able to do it, because his arms are not broken. In the first case we have to use saber, in the second case we have to use poder. The sentence "Eu sei dormir" doesn't make any sense, because there is no special training required to sleep. In this case it is always "I can sleep". "I can sleep" means that although there is a hindrance, lot of noise in the street or something like that, he can sleep. The sentence "Sei escrever" means only that in theory he can write, he has learned it, but that doesn't mean that in a concrete situation he is actually able to do it. We can say "Eu sei escrever, mas agora não posso." The meaning is that he has learned to write at school, but he has his arm broken and in the concrete situation he is unable to write.
to be allowed to
Can in English actually can be used in the sense of beeing allowed to do something. Concerning this meaning there is no difference between English and Portuguese.
can / to know
Saber means as well to know. (And in pretérito perfeito simples it means to get informed / to hear.) However it can be used as well in the sense of being able to something because it has been learned.
Same meaning as in English.
have to / must
In general we use ter que if there is a force mighty enough to impose its will. That what distinguishes have to from should. Should we use if a certain behaviour is recommended to reach a certain goal, "You should study more if you want to pass the exam", "You shouldn't do that, other people will get ofended", but in the case of should there is no force able to impose its will. The criminal has to go to prison if he has been sentenced, he not only should go to prison. The distinction made in English between have to and must is not made in Portuguese. Even in the case of an "interior necessity", "This is a movie you must see", we use ter que in Portuguese. The picture changes in conditional. In conditional ter que can express a presumption.
Should is used if a certain behaviour is recommended for moral reasons or to obtain a goal.
Conseguir means that a goal is reached or not reached after a lasting effort. The difference between "He can't do it" and "He doesn't manage to do it" is obvious. In the first case it is possible that he did not even try to do it, in the second case he made an effort and failed.
The most importante difference is between saber and poder. Sometimes both can't be translated with can in English.
no training necessary:
Eu não posso dormir.
I can't sleep.
A criança ainda não sabe escrever.
The child still can't write.
Sei dirigir um carro, mas neste momento não posso.
I can drive a car, but right know I can't.
The difference between ter que and should is the same as in English between have to and should. Have to is used if there is no alternative behaviour possible. This can have different reasons.
A stone thrown in the air has to come back to earth.
Uma pedra que se lança no ar, tem que recair sobre a terra.
A stone thrown in the air should come back to earth.
Somebody or something is mighty enough to impose its will
The tax office says that taxes have to be paid.
A fazenda pública acha que se tem que pagar impostos.
The tax office says that taxes should be paid.
An event is the logical consequence of another event
Se não se quer trabalhar, se tem que ter muito dinheiro.
If one doesn't work, on has to have a lot of money.
If one doesn't work, on should have a lot of money.
Have to is used as well in case of interior necessity.
Se você gosta de futebol, tem que vê-lo jogar.
If you like football, you have to see him playing.
In the case of interior necessity the instance that demands a certain behaviour is the subject itself. In a sentence like "I must see that" the subject assumes that he will miss something if he doesn't see it. If someone needs a break to smoke a cigarette, "I have to smoke a cigarette", there is nobody who obliges him to do that.
Should / dever is used if there is an alternative. Should / dever is used a) if there is an institution that recommends a certain behaviour, but is not mighty enough to restrict the scope of action, b) a certain behaviour is usefull to reach a goal, possible as well in condicional, c) a certain behaviour is recommended for moral reasons, d) most of all in condicional, to express a presumption.
a certain behaoviour is recommended
I should do it.
also possible in conditional form
I should do it.
a certain behaviour is recommended to reach a goal
Pequenas empresas deveriam fazer gestão de marca.
Pequenas empresas devem fazer gestão de marca.
Small companies should develop a brand.
Presumption concerning events that eventually will occur in the future can be expressed as well with the presente do indicativo.
dever to express an presumption related to the future
Ele estudou muito para seus examenes, então ele deve passar.
He has studied a lot for his examse, therefore he should pass.
Eles partiram uma hora atrás, então devem chegar em meia hora.
They have left one hour ago, therefore they should arrive in one hour.
If dever describes a presumption or a behaviour that is recommended we have to deduce from the context. In both cases we can construct with the condicional.