A dDemain, je vais aller a[à la gym./à la salle de sport ?]

"à demain" is "tomorrow" as in "see you tomorrow!" (it indicates a future prospect, usually when people are going to meet again)
simply "tomorrow" is "demain"

aller À LA gym = attend a gymnastics lesson, go somewhere to practice gymnastics
I feel like you meant "going to the gym" though, which would be "aller à la salle de sport" or "aller à la salle" (in a more colloquial way)

Alors, je mangeEnsuite, je prendrai maon petit deéjeuner et je me promeènerai avec mon chien.

"then" in the sense "after this" is "ensuite"
"alors" is "then" in the sense "at this time", so it would basically mean you'd have your breakfast in the gym

Food is "mangée" but meals are "pris" ("taken") in French => j'ai pris

"je promènerai mon chien" ("I'll walk my dog") would work too

AÀ dix heures, je vais travailler pour trois hueures et alorprès, je recontrerai avec mon amiverrai mon [ami/amie ?] pour deéjeuner et j'espeère que nous parlerons au sujet de notre projet ade voyager (pour) l'anneée prochaine.

"pour" very rarely expresses duration in French. "pendant/durant" are used instead, or nothing at all, in many cases (like here)

"rencontrer" in French mostly means meeting someone unexpectedly, for the first time or in formal circumstances. Meeting someone to have a nice time with them simply is "voir"

"amiE", if you mean a female friend or a girlfriend

A future action after "espérer" can and must take the future tense => parlerONS

"parler DE" is enough to indicate "about"

"projet de voyage", lit. "trip project" (with a "de" connecting both nouns)
Alternative => notre projet de partir en voyage l'année prochaine