Gold and Silver introduced friendship evolutions, time-of-day splits, and held item trading. Espeon and Umbreon require high friendship during day or night. Baby Pokemon like Pichu and Togepi added pre-evolution stages to existing chains. The number of evolution methods nearly doubled from Gen 1.
Walk with the Pokemon, win battles, use vitamins (Protein, Iron, etc.), and get haircuts at the Goldenrod Underground. Don't let it faint, which drops friendship. The Soothe Bell wasn't introduced until Gen 3, so the main methods are walking and battling. Friendship maxes at 255 and most evolutions need 220+.
Gold and Silver introduced the internal clock. Eevee evolves into Espeon if leveled up with high friendship during the day (4 AM to 8 PM), or into Umbreon at night (8 PM to 4 AM). Time of day also affects wild encounters and some event triggers. Make sure your Game Boy clock is set correctly.
Gen 2 added held-item trade evolutions: Metal Coat (Onix to Steelix, Scyther to Scizor), King's Rock (Poliwhirl to Politoed, Slowpoke to Slowking), Dragon Scale (Seadra to Kingdra), and Up-Grade (Porygon to Porygon2). The Pokemon must hold the item during the trade for it to work.
Seven pre-evolutions for existing Pokemon: Pichu (from Pikachu), Cleffa (from Clefairy), Igglybuff (from Jigglypuff), Togepi (hatches from the Elm egg), Tyrogue (from Hitmonlee/Hitmonchan/Hitmontop), Smoochum (from Jynx), Elekid (from Electabuzz), and Magby (from Magmar). You get them by breeding the evolved forms at the Daycare.
Tyrogue evolves at level 20 based on its Attack and Defense stats. If Attack is higher, it becomes Hitmonlee. If Defense is higher, Hitmonchan. If they're equal, Hitmontop. Manipulating IVs and EVs at low levels to hit equal stats for Hitmontop is tricky and takes some planning.