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?
Best Answer
Starting Equipment
Which class you begin with may provide better starting equipment for your eventual playstyle.
Bonus Feat
(Credit to Sdjz in comments)
If Abserd is a Variant Human, the feat gained would be limited with certain starting classes. For example, only a class with the Spellcasting or Pact Magic feature at level 1 could take Spell Sniper and a Monk could not take Medium Armor Master. It will be a long time before you have the option to get these feats another way since the earliest you could get an Ability Score Increase is level 15. As such, it could make a difference.
Unarmored Defense
While you don't have to choose either first, Unarmored Defense is mutually exclusive so if you plan on getting Monk and Barbarian levels, you should choose which Unarmored Defense you want. Barbarian uses Constitution and allows a shield while Monk uses Wisdom and does not allow a shield.