Praise be to Allah.

Firstly: 

The basic ruling with regard to hunting [translator’s note: in Arabic the same word is used to refer to hunting or fishing] is that it is permissible, except for one who is in a state of ihram (for Hajj or ‘Umrah) or one who is in the Haram zone. This applies to hunting on land; with regard to fishing and the pursuit of other aquatic creatures, it is not haraam for the one who is in ihram. Allah, may He be exalted, says (interpretation of the meaning):

“Lawful to you is what you catch from the sea and its use for food - for the benefit of yourselves and those who travel, but forbidden is what you catch on land as long as you are in a state of Ihram (for Hajj or Umrah). And fear Allah to Whom you shall be gathered back”

[al-Maa’idah 5:96]. 

If a person hunts permissible animals with a permissible intention, such as earning a living by selling them or eating them, there is nothing wrong with hunting or catching them, according to scholarly consensus. 

Similarly, if a person’s main aim, when catching fish, is permissible in principle, such as leisure and the like, nut he will make use of what he catches, such as selling it or eating it, and so on, then there is nothing wrong with him doing such things. 

Secondly: 

If the hunter or fisherman does not have any particular need for what he catches, and he is only doing it for sport (as a leisure activity or for fun), then the ruling on hunting or fishing in this case changes from permissible to disliked (makrooh). 

It says in al-Mawsoo‘ah al-Fiqhiyyah (28/115): 

If it is known that the basic principle with regard to hunting (or fishing) is that it is permissible, it cannot be deemed to be otherwise, or to be deemed makrooh, haraam or mandoob (recommended) or waajib (obligatory) except in specific cases for which there is specific evidence, as follows: 

… Hunting or fishing is makrooh if the purpose behind it is to have fun or do it as a leisure activity, because the Prophet (blessings and peace of Allah be upon him) said: “Do not take any living being as a target.”

Narrated by Muslim (1957). End quote. 

More than one of the scholars have stated that hunting and fishing in this case is makrooh. 

Al-Nafraawi al-Maaliki (may Allah have mercy on him) said: Hunting for fun is makrooh.

End quote from al-Fawaakih ad-Dawaani, 1/390 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: Hunting to meet a need is permissible; as for hunting for no purpose other than leisure or sport, it is makrooh, and if it involves wronging people by transgressing upon their crops or property, then it is haraam.

End quote from al-Fataawa al-Kubra, 5/550 

Shaykh Mansoor al-Bahooti (may Allah have mercy on him) said: It is makrooh to hunt for sport, because that is an idle pursuit. If the hunting involves wronging people by transgressing upon their crops or property, then it is haraam, because the means come under the same ruling as the ends.

End quote from Kashshaaf al-Qinaa‘, 6/213 

Ibn ‘Aabideen (may Allah have mercy on him) said: In Majma‘ al-Fataawa it says: (Hunting) is makrooh if it is for leisure.

End quote from Radd al-Muhtaar, 5/297 

Thirdly: 

If the purpose of the hunting or fishing is leisure and sport, but the person will benefit from it, by either eating, selling or giving away his catch as a gift and the like, then it no longer comes under the ruling of being makrooh, as mentioned here, and the hunting or fishing reverts to the original ruling of permissibility, because in this case it is not purely for the purpose of sport and it does not involve damage to property  or tormenting the animals caught. 

Shaykh Muhammad ibn Ibraaheem (may Allah have mercy on him) said: 

It is not prescribed to cause deal to a living being for sport, such as hunters shooting from moving vehicles, when the intention is not to eat it or feed it to others. In the hadeeth it says: “Whoever kills a sparrow unlawfully will be questioned about it.”

End quote from Fataawa wa Rasaa’il Muhammad ibn Ibraaheem Aal ash-Shaykh, 12/231 

Shaykh Ibn Baaz (may Allah have mercy on him) said: 

If it is for an Islamically-acceptable purpose, such as eating it or selling it, as in the case of those who catch buzzards, lizards and rabbits, and other permissible animals, to eat them or sell them, then there is nothing wrong with it. But if a person is hunting or fishing in order to kill the animal and leave it there, this is not right. The least that can be said about it is that it is very makrooh (disliked). No one should hunt an animal that can be eaten unless it is for a purpose, either to eat it or feed the poor with it or give it as a gift or sell it. As for hunting it for sport, that is not permissible; it is a kind of leisure or “fun” that is not appropriate on the part of the believer. It was narrated from the Prophet (blessings and peace of Allah be upon him) that he forbade hunting, except for food or to make use of (the animal caught).

End quote from the website of Shaykh Ibn Baaz

http://www.binbaz.org.sa/mat/17241

To sum up: 

Hunting or fishing, as mentioned in the question, is permissible and there is nothing wrong with it, so long as it is possible to make use of what is caught, by eating it, selling it and so on. 

And Allah knows best.