You only have to research/argue one side of the debate? Where I live, you must debate both sides at every tournament... which I think is great because it gives you a more balanced look at all issues (which really helps one in life so one doesn't become a one-track-thinking, stubborn person) and prevents someone from getting hit when the questions is worded such that it is slightly one sided.
Oh, and just before I begin... who decided your debate topic anyway? Generally they are phrased in an affirmative sense so it isn't confusing to the audience. What I'm saying is, it would've been better to phrase it as "Resolved that government aid programs offer more benefits than problems" so that you don't have to think backwards... it is sometimes hard for the audience to understand when you are the "con" side and arguing that something is "good". Normally, "con" argues that something is "bad". This sounds minor, but you'd be surprised how easily the audience gets confused, especially if you refer to the other team as the "pro" team when they are saying something should be scrapped. Don't PROponents advocate things, not say they should be scrapped? Be careful to make sure you yourself don't get confused (especially if you have to argue both sides at one point... I've seen people start arguing the wrong side, especially during a tricky cross examination).
Sorry, I digress...
Now, I don't know how much debating you do, but the first thing you should do is to make sure that the definitions are clearly defined. When I debate, it is the affirmative (pro) side that defines the terms, however, if they do not, you are free to define the terms as you like. If affirmative does define the terms, be prepared for however they define them... if they define "Government Aid Programs" as "any form of wealth or service distribution or redistribution by any government" then the debate will be different than if they define it as "welfare as provided by the federal government".
Be careful, they might throw you for a loop and define government aid programs as "financial aid provided by the federal government to foreign developing nations" -- that is indeed government aid and would not seem to be a "squirreling" of the definition (they could then argue that the dictator normally just spends the money and the people never see it). While definitions are often thought to be unimportant, lawyer-ish and lousy semantic bickering, it nevertheless remains and integral part of the debate. So be prepared for it.
Assuming a conventional, broad definition, there are several ways you can approach this. What I often like to do is research 3 solid case studies where the government aid programs were vital in keeping society running smoothly. FDR's stuff is probably good, if you want (depending on how the terms are defined, again), you can also pick an international example (most likely your opponents will not be familiar with it, therefore they will have a harder time attacking... the flip side is of course it may not seem as important or compelling to your audience, so you have to explain it well), or a state example, or a local example (so that it doesn't look like you are reciting examples from textbooks, and also because these issues tend to be more pressing). If you want, you can also bury them with lists of examples where government aid did good... don't get too specific with each one, just list so many that they'll have a hard time covering them all... and if they don't cover any single one of them then you can accuse them of dodging your arguments.
And don't forget that (in most cases, check to be sure) the burden of proof lies with the affirmative (pro) side. So theoretically, you don't even have to present your own arguments as long as you refute theirs. If they say that <whatever aid program> was a failure, and you successfully argue otherwise, then their case is already dead.
Sorry I'm not giving you too much specific help, but as has been mentioned, this topic is very, very broad. What you argue is soooo dependent on how "government aid" is defined. It could be anything from foreign aid, to compensation to families of people who died in military accidents, to social welfare programs, to any government service. To be honest, without the topic being more specific, the winner will probably end up being the person who can manipulate the audience the best (that's even true when the topic is specific, but even more so here). If you can narrow it down, I could probably provide more topic-specific help.