There really was not that much of a choice but for Ukraine to give these weapons up. There was a widespread international consensus, not just limited to Russia and the US, that Ukraine should not be allowed to join the nuclear club. There were substantial fears at the time that the poverty of the nuclear inheritor states would mean they did not have the resources to provide adequate safety to prevent the theft or illegal sale of nuclear materials. Additionally, having to deal with a new nuclear power, which was by some counts the third-largest nuclear power after the break-up of the USSR, would further complicate existing nuclear treaties and put the skids on the trend of international arms agreements limiting nuclear weapons. American and European economic assistance became predicated upon removal of these weapons.

But besides international opinion being largely against Ukrainian nuclear capacity, there was a strong domestic push within Ukraine in favor of nuclear disarmament as well. It was not just the Europeans and the US who made the connection between aid and disarmament. The Rada and Ukrainian leadership also saw the stockpile as a useful bargaining chip to wrest concessions out of the West and Russia. Getting rid of the arsenal bought Ukraine a degree of international respectability, agreements on Ukrainian security, and some material benefits.

The arsenal itself was not really a viable defense option for the country. Although the stockpile of ex-Soviet nuclear weapons was quite vast, the technical and maintenance infrastructure of Ukraine was rudimentary. Keeping these weapons operational over the long-term would have been expensive as the Ukrainians would have had to build up this support infrastructure from scratch to recondition weapons whose shelf-life was quite short. Cannibalization for spare parts and paring down the stockpile might have been viable over the short-term, but retaining these weapons was a long-term fiscal impossibility unless the Rada was willing to pass extreme budgets. Moreover, a large chunk of the stockpile were long-range strategic weapons which would be difficult to repurpose against Russia. By the same token, any seizure of the weapons against the teeth of international opinion would have meant that the Ukrainians would not have the security keys or safeguards for these weapons, rendering the weapons as they existed functionally inert. A hypothetical Ukrainian nuclear stockpile might have to use the nuclear material in the Soviet warheads to fashion their own indigenous bombs which would be cruder and possibly be duds without prior nuclear testing.

The international consensus for a non-nuclear Ukraine also meant keeping these weapons of mass destruction would have placed the country into the position of a rogue state. In the immediate aftermath of the Gulf War, this was not enviable company to keep, especially for a newly independent nation seeking legitimacy.

Pulling together relevant parts of this answer and this answer that I previously wrote.

The Soviet military continued after 1991 as a transitional Commonwealth of Independent States military theoretically under joint command of its member states. Russia had declared itself to be the legal successor to the USSR, assuming its UN seat, its treaty obligations, its foreign debt, and control of its nuclear arsenal. The Alma Ata Protocols in December 1991 recognized that the Russian President held command and control of nuclear forces and was to use them in consultation with other members of the Commonwealth of Independent States, most notably the three other "nuclear" states that had weapons deployed on their territories (Belarus, Ukraine and Kazakhstan). Hotlines were established between these capitals and the Kremlin, and when Gorbachev resigned on December 25, 1991, he transferred his nuclear codes (kept in the "cheget" nuclear briefcase) to Russian President Boris Yeltsin.

A quick word on the nuclear codes and nuclear forces. The codes used a "triple key" system: the Soviet/Russian President had to transmit his code to the Minister of Defense, who had to transfer their code to the Chief of Staff. This controlled strategic missiles commanded by the Strategic Rocket Forces (a separate military branch), submarine missiles under control of the navy, and airborne weapons controlled by the Air Force. In late 1991, Gorbachev unified all nuclear weapons under one command, but this was reversed later in 1992. In April 1992, Russia created its own Ministry of Defense, which shared personnel (and nuclear codes) with the older joint military, which uneasily coexisted and shrank in importance before being dissolved in June 1993. So for a while there were technically two command and control systems over the former Soviet arsenal. After the dissolution of the joint CIS command, this control was all folded into the Russian Ministry of Defense structure.

Generally speaking, Belarus and Kazakhstan were more or less fine with de facto Russian control of nuclear weapons on their territory. Ukraine, less so. From late 1992, Ukraine had custodial control of nuclear warheads on its territory, and set up its own embryonic command and control system, as well as protocols to order military staff on its territory to not comply with launch orders that the Ukrainian President did not countersign. Ukrainian control went so far that in 1993 the Ukrainian military removed and transported warheads from missiles, inching closer to setting up its own active control of nuclear weapons. This confusing situation was only ultimately resolved by the December 5, 1994 Budapest Memorandum, under which Belarus, Ukraine and Kazakhstan agreed to the transfer of all nuclear warheads to Russian territory, which was completed by 1996.

Budapest Memorandum of December 5, 1994 was agreed between Ukraine and Russia, the UK and the US (very technically it's actually three memorandums with identical terms between Ukraine and each of the other countries). A copy submitted to the UN General Assembly can be found here. France and China also gave separate unilateral security assurances to Ukraine.

Memorandum isn't a formal treaty - it's basically bilateral assurances. The closest to a formal pledge on Ukraine is Article 2:

"The Russian Federation, the United Kingdom of Great Britain and Northern Ireland and the United States of America reaffirm their obligation to refrain from the threat or use of force against the territorial integrity or political independence of Ukraine, and that none of their weapons will ever be used against Ukraine except in self-defence or otherwise in accordance with the Charter of the United Nations"

There aren't specific terms or obligations listed should this be violated though, except in Article 4, in which case the US, UK and Russia pledge to seek immediate UNSC action should a country use nuclear weapons against Ukraine, and Article 6 (that the countries would consult "in the event a situation arises that raises a question concerning these commitments."

Strictly speaking, the treaties that Ukraine was party to that actually governed its security and its nuclear weapons are referenced in the memorandum, namely the Treaty on the Non-Proliferation of Nuclear Weapons (which Ukraine joined after signing the Memorandum), the Helsinki Final Act of 1975 (which Ukraine signed in 1992) and the UN Charter (which Ukraine as the Ukrainian SSR signed in 1945 as a Founding UN Member).

The nuclear arsenal that was on Ukrainian territory consisted of the following: 130 SS-19 and 46 SS-24 intercontinental ballistic missiles (ICBMs) with 1,240 warheads, and 44 Tupolev-95 and Tupolev-160 strategic bombers (with 1,081 nuclear cruise missiles). There apparently were still tactical nuclear warheads on Ukrainian territory as well - the USSR basically kept tactical nukes in local armories, and started relocating them to Russia proper in the last months of 1991. The last ones were out of Ukraine in early 1992. So the negotiations were always specifically around the strategic weapons and delivery systems.

Ukraine started negotiating with Russia over the fate of the strategic nukes starting in 1992 (when Russia relatively quickly worked out deals with Belarus and Kazakhstan). Part of why Ukraine's deal took longer and ultimately involved the US was because Ukraine was driving a harder bargain. It did want assurances over its sovereignty and territorial integrity, but also specifically had more cost-conscious concerns as well. It wanted to make sure it was fully compensated for the value of the Highly Enriched Uranium in the warheads, and also that as few costs for eliminating nuclear weapons and infrastructure as possible would actually be paid by Ukraine. So even from the Ukrainian perspective it wasn't so much a question of "if" as much as "when, and at what price".

The Russian-Ukrainian bilateral negotiations carried on and in theory worked all this out by September 1993, but that particular deal (part of the Massandra Summit) fell apart pretty much as soon as it was agreed to, and then the US stepped in more actively in trilateral negotiations., and even by January 1994 all three parties had more or less set out the general terms for the weapons removal, as codified by the end of the year. For what it's worth the last warhead was transported to Russia by June 1, 1996 and the last nuclear delivery vehicle (an SS-24 missile silo) was dismantled in 2001.