If you exclude adjectives with stems ending in к,г,х,ш, щ, ж, ч, ц, there are two really visible patterns: soft-stem and hard-stem. For example, let's take "белый" and "синий" and watch their endings: Nom: белый/синий , белая/синяя, белое/синее, белые/синие Gen: белого/синего, белой/синей, белых/синих Prep: о белом/синем, о белой/синей, о белых/синих ("ый" should be ОЙ when stressed: "дорогой") As you can see, hard-stemmed adjectives use "hard" vowels and soft-stemmed use iotated vowels. You may memorize the "soft" endings by noting that a iotated conterpart of "о" is "ё". However,"ё" cannot be unstressed (and Russian adjectives use a fixed stress in their full form!), so you use "е" instead.

Летний is an adjective. In Russian there is declension of adjectives and nouns. You have to determine gender of the adjective ( in your sentence it is neuter gender, because the noun is neuter gender) and learn changing of the endings. Nominative case: кто?, что? -летнее путешествие Genitive: Кого?, чего? - летнего путешествия Dative: кому?, чему? - летнему путешествию Accusative: кого? что? - летнее путешествие Instrumental: кем?, чем? - летним путешествием Prepositional: о ком?, о чём? - о летнем путешествии.

Adjectives depend on nouns, to which adj is related. "Путешествии" - is noun in male gender and Ablative case, therefore, adj "летнем" has male gender and Ablative case, the ending to this is "ем"

The base of the word летний "летн" ends up with a soft consonant. It must remain soft in all the cases. But it's just my opinion.