Going on honeymoon trips is something which was neither known nor practiced by earlier Muslims. It also is not a part of the non-Muslim (ritual/worship) practices and therefore it has not per se the status of bid'ah.
Note that going on 'umrah after marriage also is not a Muslim practice that one may find any evidences for in the early days of Islam. So if people introduced this it might become a bid'ah.
Scholars have different opinion and objections on the so called honey moon here just some of them:
- it is a waste of time
- it is a waste of money
- visiting non-Muslim countries or countries where gender mixing and immorality is practiced is discouraged.
So if one could avoid these things or limit them to an adequate manner such a trip isn't prohibited and might be lawful. Some scholars give the advice to visit the mosque of the prophet () or (historical) mosques in general etc.. So this trip should also have some parts that reminds spouses of Allah and get spouses together in their worship of Allah.
The scholar 'Abdarrahman as-S-heem said it is permissible for newly-married spouses as this is a way to get them closer together nevertheless the husband (or in general for trips the guardian of a lady )should consider avoiding to take his wife (or a lady/girl) to places or countries were gender mixing etc. is allowed. He also considered honeymoon as a custom that has no link to religious practices and therefore falls under the kind of customs that must be treated separately based on the circumstances and kind (no general verdict). And one shouldn't take the word literally so a short trip might be sufficient and it is even better not to call it honeymoon شهر العسل (shahr al-'Asal in Arabic means literally "month of honey"). The full fatwa was quoted in this Arabic thread.
Here a part of a fatwa on honey moon of Sheikh ibn 'Othaymeen:
With regard to what is called the “honeymoon”, it is even worse and more reprehensible, because it is an imitation of the non-Muslims and it is a great waste of money. It also leads to neglect of many religious matters, especially if this time is spent in a non-Muslim country, after which they come back with habits and customs that are harmful to them and to their society. These are matters which are dangerous to the ummah. But if the man were to travel with his wife to do ‘Umrah or to visit Madinah, there is nothing wrong with that, in sha Allah. (Source islamqa #171265)
In another fatwa Sheikh ibn 'Othaymeen actually said something which sounds somewhat different when asked how about 'Umrah instead of or as honeymoon:
In the following I will translate from Arabic, -if not indicated otherwise- the translation is mine take it with the necessary care!
نقول: هذا حسن وغير حسن؛ لأن الظاهر أن أصله مأخوذ من غير المسلمين؛ لأننا ما عهدنا هذا في أزمان العلماء السابقين، ولا في عهد السلف، ولا تكلم عليها أهل العلم، فيكون هذا متلقى من غير المسلمين، هذا من وجه.
We say this if fine and not fine, because the apparent is that the source of this is taken from non-Muslims, as we were not used to this practice at the time of earlier scholar, nor in that of the salaf, nor did any of the people of knowledge mention it. So this is taken from non-Muslims this is the first objection.
ومن وجه آخر أخشى أنه إذا طال بالناس زمان أن يجعلوا الزواج سبباً لمشروعية العمرة، ثم يُقال: يسن لكل من تزوج أن يعتمر! فنحدث للعبادة سبباً غير شرعي وهذا مشكل؛ لأن الناس إذا طال بهم الزمن تتغير الأحوال وينسى الأول، فلهذا نقول: أجعل شهر العسل في حجرتك، في بيتك، واجعل العسل دهراً لا شهراً، واحمد الله على العافية.اهــ
The second objection I fear that if people went forth in this practice they might one day consider marriage being a reason to perform 'Umrah. And people would say it is sunnah for all those who get married to perform 'umrah! And we would create for a worship an illegal reason/condition and that is an issue! As people whn time passes change their conditions and what was earlier might be forgotten. So we say let the honeymoon be in your room, in your house and let the month be a year (or the rest of your life) not a single month. And thanks to Allah for the wellbeing (End quote). (Source: This Arabic thread quoting fatwas from Sheikh Saalih al-Fawzabn and ibn Baz too)
Finally let me add a sunnah that one may follow when newly married:
It is the Prophet's tradition that if someone marries a virgin and he has already a matron wife then he should stay for seven days with her (the virgin) and then by turns; and if someone marries a matron and he has already a virgin wife then he should stay with her (the matron) for three days, and then by turns. (Sahih al-Bukhari)
'Abd al-Malik b. Abu Bakr b. Abd al-Rahman b. al-Harith b. Hisham reported on the authority of his father from Umm Salama (Allah be pleased with her) that when Allah's Messenger (ﷺ) married Umm Salama, he stayed with her for three nights, and said:
There is no lack of estimation on the part of your husband for you. If you wish I can stay with you for a week, but in case I stay with you for a week, then I shall have to stay for a week with all my wives. (Sahih Muslim)
so if you planed a "honeymoon" you may stay with a virgin wife seven days and with a matron three days.