I'm building the character Abserd, who has 1-level of all classes. He really is a useless character, got kicked out of every guild he ever joined.
There are some specific benefits to starting with specific characters, in HP, Saves, and proficiencies. However, since we are taking levels in all classes, we can just compare what we gain, that we wouldn't gain otherwise. For example, since at least 2 classes give Shield proficiency, it's irrelevant if the starting class does so. Since dipping into Ranger gives you 1 skill anyway, starting with Ranger doesn't give more skill proficiencies than starting into Fighter.
Here's what I cooked up. Abserd, regardless of his starting class, will have proficiency in simple and martial weapons, shields, and light and medium armor. The table below shows benefits specific to starting with each specific class.
\begin{array}{|c|c|c|c|c|c|} \hline & \textbf{HP} & \textbf{Skills} & \textbf{Armor} & \textbf{Tools} & \textbf{Saves} \\ \hline \textbf{Barbarian} & 5 & & \text{} & \text{} & \text{STR/CON} \\ \hline \textbf{Bard} & 3 & & \text{} & \text{1 Musical} & \text{DEX/CHA} \\ \hline \textbf{Cleric} & 3 & & \textbf{All*} & \text{} & \text{WIS/CHA} \\ \hline \textbf{Druid} & 3 & & \text{} & \text{Herb. Kit} & \text{INT/WIS} \\ \hline \textbf{Fighter} & 4 & & \textbf{All} & \text{} & \text{STR/CON} \\ \hline \textbf{Monk} & 3 & & \text{} & \text{1 Artisan's} & \text{STR/DEX} \\ \hline \textbf{Paladin} & 4 & & \textbf{All} & \text{} & \text{WIS/CHA} \\ \hline \textbf{Ranger} & 4 & & \text{} & \text{} & \text{STR/DEX} \\ \hline \textbf{Rogue} & 3 & 1 & \text{} & \text{} & \text{DEX/INT} \\ \hline \textbf{Sorcerer} & 2 & & \text{} & \text{} & \text{CON/CHA} \\ \hline \textbf{Warlock} & 3 & & \text{} & \text{} & \text{WIS/CHA} \\ \hline \textbf{Wizard} & 2 & & \text{} & \text{} & \text{INT/WIS} \\ \hline \end{array}
:* Depends on sub-class
Barbarians have the most HP of all starters, and Rogues have 1 additional skill proficiency. If you want proficiency in Heavy Armor, you need to start with Fighter or Paladin (unless you pick some specific Cleric Domain when you dip cleric). You can also get an additional proficiency with an instrument/tool if you start with Druid/Bard/Monk.
so, TL;DR, aside from these bonuses, and the Saving Throw proficiencies, are there any other benefits to starting with a specific class?