J'ai commencé à [prévoir/planifier un?] des vacances.

"vacances" is always pluralized in this sense is French: "une vacance" means there's something (like a position for instance) that is vacant
"planifier" is much more concrete and developped than "prévoir". If it's a mere idea for now, "prévoir" would be the word

Plus tôt cette année, je'ai voyagerais àé en Tasmaniae.

Honestly, just saying "cette année" would sound a bit more French (the fact it was "earlier" is already made clear by the past tense)

Unique action which is achieved, main action = passé composé => j'ai voyagé ("voyagerais" is conditional present)

The preposition rules concerning islands are somehow erratic in French, but it is "EN Tasmanie" (maybe because "Tasmanie" is also considered a State)

C'était amazementmerveilleux !

One would use an adjective is => merveilleux, magnifique, superbe, génial

Je n'avétais jamais été àallée en Tasmaniae avant donc c'était une expérience complètement nouvelle.

"être" in the sense of "aller" is clumsy and not recommended at all, especially in a text

J'ai resté avecséjourné mes cousine(e)s et [ils ont me/elles ?] m'ont montré leurses endroits préférés.

"cousins/ils" if they were male cousins or males and females together, "cousinEs" and "elles" if they were only females

Nous mangeons unavons mangé beaucoup de bon nourriture et regardionss plats et avons vu la campagne.

See above for the passé composé

to see = voir
"regarder" rather means "to look at"

Pour mes prochaines vacances, j'irai/je vais allerais à l'étranger.

"allerai" doesn't exist; the indicative future and conditional present forms are built on the root "ir-" => j'irai, tu iras, il/elle ira, nous irons, vous irez, ils iront (future)
"je vais aller" is also possible if you're referring to a near future

Je ne sais pas où, mais je pense ouque ce sera soit la Nouvelle-Zélande ousoit Singapour [et/ou bien?] Japon.

"Singapour et le Japon" means "either New Zealand one one side, or Singapore and Japan together on the other side"
If it's just one of the three alone, you'd have to use "ou bien"

Je pense que la Nouvelle-Zélande serait une bonne destination parce que la campagne est donctellement jolie et je voudrais faire des randonnées.

If you meant "so pretty", it is the so indicating a degree of something, so "tellement" or "si" in French
"donc" is the "so" indicating a consequence

Singapour et le Japon seraient chouettes à visiter, parce que leur nourriture serait amazementcuisine/gastronomie est délicieuse/succulente/extra et il serait intéressant d'explorer leur(s) cultures(s).

"LE Japon" for the same reason as "LA Nouvelle-Zélande"

The sum of all dishes belonging to a given culture/country is called "la cuisine" or "la gastronomie" (once again, it's a much better than the very general "nourriture")

"extra" is more spoken and colloquial than the other alternatives, but all of them would be right

The French language would consider each has one culture and therefore use the singular, but the plural is right too