The group, which is 80% owned by the taxpayer, said pre-tax profits rose to £1.64 billion from £826 million in the same period last year, but boss Ross McEwan said there are still "plenty of issues from the past to reckon with".
There were no new hits to cover past scandals or litigation, or major provisions such as the £4.8 billion hit it recently took to create a "bad bank" where it could hive off toxic assets.
But RBS admitted that it would continue to be haunted by misdemeanours of previous years, which have seen the bank shell out billions of pounds over issues such as payment protection insurance (PPI) mis-selling.
It said: "The ongoing conduct and regulatory investigations and litigation continue to create challenges and uncertainties for RBS, as for other banks. The timing and amounts of any further settlements or redress remain uncertain."
RBS said it had seen a modest revival in lending volumes during the quarter, with improvements in UK retail and business banking while income from its markets business was lower as it shrunk its balance sheet. Costs also fell.
The trading update was the first since the group announced that it had tumbled to an £8.2 billion loss for 2013 and launched a mammoth overhaul to slash costs by £5 billion within three years.
Chief executive Mr McEwan said the latest figures showed the "great job" it could achieve while in a "steady state".
"But we still have a lot of work to do and plenty of issues from the past to reckon with," he added.
"Everyone at RBS is focused squarely on doing everything we can to earn the trust of our customers and in the process change the banking sector for the benefit of the UK."
Shares surged by 12%, adding more than £2 billion to RBS's market value.
RBS said gross new mortgage lending of £4.4 billion in the first quarter included more than 4,700 approvals "assisting young people and families to buy their first home" through the Government's Help to Buy scheme.
It said it was on course to achieve targets for improving the level of capital on its balance sheet - a measure lenders have been ordered to take by regulators in the wake of the financial crisis, to bolster their ability to absorb shocks.
Meanwhile, its troubled Ulster Bank subsidiary reported its first quarterly operating profit since 2009. It earned £17 million in the period, compared with a £164 million loss in the same quarter last year and a £996 million loss in the fourth quarter of 2013.
RBS has reduced staff numbers by 6,300 compared with the same period last year, making expenses £174 million lower.
It also said it was shaking up computer systems for NatWest and Ulster Bank as part of a wider £750 million three-year programme to improve "safety, security and resilience" - following a series of IT failures across the group in recent years.
RBS said it was seeing increasing economic confidence in some areas and expected a modest increase in margin for the remainder of the year. But it is likely to be hit by costs for as part of its restructuring plan that are "considerably higher".
Mr McEwan pledged to do what it takes to hold on to highly-paid bankers who would be affected by a Government decision to block it from paying bonuses twice the size of salaries.
Asked whether fixed salaries for those employees would be raised to compensate, he said: "I just want to make sure that we are in a position to pay those people so we can retain them."
The quarterly trading update noted that all the group's major competitors planned to be able to pay bonuses at twice the rate of salary - which must be approved by shareholders under new European rules. RBS is limited to paying 100% bonuses.
It said this created a "commercial and prudential risk which it must try to mitigate within the framework of a 1:1 fixed to variable compensation ratio".
Mr McEwan pointed out that the Government was publicly opposed to the EU bonus cap rule but acknowledged that pay was an "emotive issue".
He said RBS was a "back marker" on pay, adding: "It is important that we stay in the pack rather than being out on our own as we do change this business."
Mr McEwan said the cap affected a small number of people involved in its international markets division, in its restructuring, and in the US.
He said: "We are not going to pretend that this is ideal.
"Not having the flexibility does involve an element of risk for us but it is a risk that I as chief executive am going to have to manage and we are having to make some changes to those people to make sure we hold on to them."
Mr McEwan also said RBS believed it had a strong defence against a major legal claim against it over the group's 2008 rights issue.
"These things will be seen out in court as opposed to a settlement," he said.
Shore Capital analyst Garry Greenwood said: "While 'one swallow doesn't make a summer', we believe that the Q1 2014 outcome represents an excellent start to the year for the group following recent disappointment.
"That said, there is still plenty of work to do before RBS gets back to being a 'normal' bank."
Richard Hunter, head of equities at Hargreaves Lansdown stockbrokers, said: "It seems that these numbers represent the calm before the storm. The initial share price reaction seems tinged with some relief.
"The quarterly performance itself is respectable - a significant reduction in impairments, costs down, an improvement in underlying profit and a relatively robust capital cushion.
"Much further out, the streamlining of the bank and a following wind may enable the resumption of a dividend and the removal of the stultifying government stake.
"However, in the meantime, the bank is still in the midst of dealing with its legacy issues, and has warned that there will be a considerable rise in costs as the year progresses, driven both by the restructure and the possibility of further regulatory fines."