Would multiple t-tests be appropriate I have reaction time data from four different age groups, and I am hoping to prove that reaction time improves with age. I know that ANOVA would be preferred to determine if age has an effect, but I am not looking to simply prove that or to compare each group to every other group. Instead, I only want to compare Group 1 to Group 2, Group 2 to Group 3, and Group 3 to Group 4 (with the hope that Group 4 mean is less than Group 3 mean, Group 3 is less than Group 2, and Group 2 is less than Group 1).
Would three t-tests be appropriate here, or would a different method be preferable?
 A: You could do a series of one sided tests for such a directional alternative, though it's also possible to test such hypotheses in a single test.
There are several nonparametric tests - e.g. the Jonckheere-Terpstra and Match test for ordered alternatives (found, for example, in Neave and Worthington's, 1988 book Distribution-Free Tests.).
There are also parametric approaches for testing for trend. The book by Barlow, Bartholemew, Bremner and Brunk (1972) (Statistical Inference Under Order Restrictions) discusses some, though there are numerous more recent papers, some based on isotonic regression approaches, and some on linear-score tests for example.
There's also Bayesian approaches. For example Gelfand et al (1990) and  Gelfand et al (1992) discuss using MCMC for an ordered alternatives problem (the second in a two-way situation).  
Gelfand, Hills, Racine-Poon, and Smith, (1990),
Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling,
Journal of the American Statistical Association, 85:412 (Dec.), pp. 972-985
Gelfand, A.E., Smith, A.F. & Lee, T. (1992),
Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling,
Journal of the American Statistical Association, 87, pp. 523–532
Of those, I'd be inclined to just do a Jonckheere-Terpstra, simply because its probably the most readily doable without going to extra effort; it can be found in quite a few packages, though some of the parametric trend tests may be available to you as well.
