The European Council of Foreign Relations' website is hosting a roundtable of views from around the region on the Syrian conflict that's very much worth reading to get a hold of the complexity of its regional dimension.
Hassan Hassan starts off with what the Gulf states want:
The potential demise of the pro-Iranian regime in Damascus offers the Gulf states the possibility of extending their regional influence. Saudi Arabia and Qatar, in particular, believe that a friendly regime in Syria will give them influence over Shia-dominated Baghdad, over whom they have had little sway, but which is seen as a critical player in the regional balance of power. Iraq’s post-2003 alliance with Iran is perceived as one of the key reasons. for Tehran’s growing regional influence over the past decade. A Sunni state in Syria could serve to strengthen currently marginalised Iraqi Sunni forces, giving them – and their Gulf backers – greater influence in Baghdad. At the same time, regime change in Damascus would help the Gulf states bolster their standing in Lebanon, already economically dependent on the Gulf, by strengthening pro-Sunni Gulf actors at the expense of the dominant pro-Assad Hezbollah movement. For the Gulf States, the Syria conflict is thus a critical battle for control of a key pivot state in the region. Drawing Damascus away from the Iranian camp is seen as a way of cementing broader regional influence in the Levant, and reestablishing the more favorable regional balance of power that they lost following the US occupation of Iraq in 2003.
And from Haydar al-Khoei's piece on the view from Iraq:
A diplomatic incident in Damascus sheds some light on how events in Syria are being seen by Baghdad. In the summer of 2011, the Qatari ambassador to Syria invited several Arab ambassadors as well as the Syrian foreign minister to his residence. Whilst sitting around the dinner table the Iraqi ambassador remarked, “The same people who conspired against Iraq are now conspiring against Syria.” This enraged the Saudi Arabian ambassador, who responded, “I dare you to name them. I dare you!” The Syrian foreign minister attempted to calm the situation by saying, “The Iraqi ambassador is referring to al-Qaeda and the Salafis, not Saudi Arabia,” but the undertone of the message was clear.
There's much more to read there - check it out. ECFR's latest report on Syria stakes out the wise position, in my view, that any alternative to diplomacy would be disastrous.