The Taiping certainly did not capture cities unmolested: the siege of Changsha was a failure in which one of the main Taiping leaders, Xiao Chaogui, was killed in action. The Taiping had also suffered an earlier reverse at the Suoyi Ford, where a militia commander blocked and destroyed the Taiping river flotilla and supposedly killed or captured 10,000 Taiping followers. But it is true that the Taiping were generally quite successful to begin with, and that deserves some degree of explanation. This very recent answer discusses the early Taiping, but of course we can't just extrapolate that straight onto the next two years.

One factor is just the vicissitudes of military fortune for the Qing: like the Communists in 1934, the Taiping were twice able to break out of Qing attempts at encircling them. Encirclements are nice if you can assemble the forces to cover every point effectively, but they also thin out your lines, and unless you are decisively superior numerically, the encircled enemy might still be able to concentrate enough force at one or more opportune points to break out, as the Taiping did.

Related to that was limitations in Qing manpower: the Green Standard Army was basically a mixture of armed constabulary and local defence force, and the proportion that could be spared beyond a particular province or even locale varied based on circumstances. During the Opium War for instance, no province besides Guangdong and Zhejiang, where the war itself was fought, spared more than 37% of its troops towards the war effort over the course of some three years according to Mao Haijian. The real mobile force was supposed to be the Banner armies, but one issue was that relatively few Banner garrisons existed in the upper Yangtze region to be mobilised, and these were generally small. To put some hard numbers on it, if we look at the ideal sizes of Qing forces in the early provinces of Taiping activity, they are as follows:

Province	'Garrison' Green Standards	Other Green Standards	Banner Soldiers	Total
Guangxi	12808	10305	0	23113
Hunan	16477	10045	0	26522
Hubei	14202	8359	4963	27524
Anhui	5861	2435	0	8296
Jiangsu	25390	13456	5159	44005

Figures are for 1825, from Thomas Wade's The Chinese Army (1851).

As you can see, the Qing had no more than 50,000 troops in any one province, and in all cases over 50% of the troops were designated as garrison forces and unlikely to be particularly viable to mobilise. According to Mao Haijian, the cavalry-infantry-garrison distinction was mainly of pay grade rather than function, but even if so, the numbers we're looking at aren't particularly vast compared to the Taiping army, which by March 1853 numbered 500,000 total people per Jen (1973), but I suspect this is a number that includes the entire civilian following, and the number of actual troops was probably closer to 100,000 – they were at 70,000 fighting men (and perhaps women) as of December 1852, according to Elleman (2001), who otherwise repeats Jen's 500,000 figure for 1853. The Qing did occasionally manage to achieve significant concentrations of force: the Changsha defenders that held off the Taiping eventually numbered some 50,000, for instance. But if the Taiping kept moving, then they would keep running into relatively limited pockets of defenders.

It's also worth noting that after several attempts to defeat the Taiping in the field, the garrison forces at Nanjing had been severely drained. According to Jen Yu-Wen, by the time the Taiping arrived, the 4000 to 5000-strong Banner troops were supporting a mere 1,800 Green Standard regulars, and a little over 8500 mercenary and militia troops hired on short notice. While having substantial defences helped, in the end the Taiping simply had far more force to concentrate against the Qing defenders.

The Taiping were certainly lucky in that they had already built up a sizeable following over 1844-50, and that Guangxi was just not a particularly militarised province. As a result, they started out in a region without much means to prevent the growth of an as-yet-not-rebelling force, and then moved on to several regions that didn't have that much more military manpower either. When they were confronted either with an effective stratagem as at Suoyi Ford, or a comparable concentration of troops as at Changsha, the Taiping didn't have much chance, but they simply had a highly concentrated, mobile field army that could otherwise easily mop up the small garrison forces the Qing normally had in place. That isn't to say the Qing were inactive in responding to the Taiping by any means, but it is to say that they lacked both the operational initiative and the initial forces to put up a firm resistance outside a small handful of key junctures.