What is allowed on the Sabbath to me is like anyother ethical question for the christian it is always framed withen a general scheme or categories of ethical judgements: good things, bad things, and gray things. Good things would be like praying, going to church, singing hymns, worship, bible study. Bad things would be like working unecessaraly, not going to church other than for being sick or something, worshipping in another religion. Gray things are those activities that are not so easily definable like your going on a walk example. These are things that honest christians can honestly disagree over. It is not to say that they neither right or wrong in an ultimate sense its just that they are not so easily definable one way or another.
In my opinion the biggest shame over this issue is not so much what people do on Sunday but the fact that they don't even think about it. You may do something questionable on the Sabbath but if you honestly wrestled with it, sought Godly advice, studyed the bible over it than you are at least a step ahead of most people. We ought to be thinking about whether or not we should do something on Sabbath prior to doing it, that at least shows that our perspective is in the right spot.
So you must decide for yourself whether or not you should do a particuler activity on the Sabbath, God alone is Lord over the consceince so my guess is He'll convict you of those things that He doesn't want you in particulier to do on the Sabbath. Also what is your church's opinion on the matter. I'm not saying that you need to to get an OK for everything you want to do on the Sabbath, but you should know your church's general position on the matter and bring that into your decision proccess, after all you took vows and everything.