I was reading that Stalin initially supported Israel and the USSR was one of the first nations to recognize it.

Indeed! He supported the partition plan, and the USSR was the first nation to grant Israel de jure recognition (within 3 days), as opposed to de facto recognition (less powerful, was first given by the US within 11 minutes).

In fact, the USSR apparently sold a goodly amount of weapons and equipment to the Israelis

Also true! The USSR sold weapons to Israel...well, a Soviet proxy did, Czechoslovakia in fact. These weapons were a huge boon to the new state in fighting the Arab nations in 1948, and arguably are one of the most crucial reasons they were able to succeed (among other factors, like more manpower and better coordination/morale).

Then, in 1949, Stalin changed his mind and decided to oppose Israel. Why? Wouldn't it have made more sense for the Soviets to continue to court the Israelis, many of whom were Russians or Eastern Europeans?

A good source for this is "SOVIET - ISRAELI RELATIONS" by Klinghoffer. Labour Zionists, the group that led Israel's coalitions until 1977, were initially close to socialist and had warm connections with the Soviets. However, Cold War rivalries and systemic differences soon got in the way. Israel, a democracy, tilted towards the West. Western Jewry and Western states (especially France in the 50's) became the main supporters and aid-providers of the new state, further tilting it West. Fear of Zionism spreading through its own citizens led to the Soviets banning almost all emigration to Israel (5 Jews left legally for Israel), and the anti-Semitism of the cultural purges under Stalin not long before Israel's establishment was not forgotten. And by 1953, when the "Doctor's Plot" supposedly found Jews working for the CIA and Israel to kill high-ranking Soviets, and when the resulting bombing of the Soviet legation by Israeli militants that same year occurred, relations took a nosedive.

In reality, the Soviet state gravitated towards the more populous and dictatorial Arab States, and as Cold War lines solidified and Israel worked more closely with the West and the Soviets with the Arab states, the Israelis became firmly entrenched with the Western powers. By the time the 1967 War rolled around, Israel was fighting Soviet patron states, who had been supplied by Soviet weapons as Israel was in 1948. The 1967 War even prompted an outbreak of anti-Zionist (which was a cloak for anti-Semitic, and anti-opposition) actions by the Soviets and their proxy states in the Soviet bloc, resulting in arrests of Jews and "pro-Israelis", as well as the flight of Polish Jews, of whom there weren't all that many left to begin with (relative to pre-WWII, of course).