You could create your own .xml animation files to fade in a new Activity and fade out the current Activity:
fade_in.xml
<?xml version="1.0" encoding="utf-8"?>
<alpha xmlns:android="http://schemas.android.com/apk/res/android"
           android:interpolator="@android:anim/accelerate_interpolator"
           android:fromAlpha="0.0" android:toAlpha="1.0"
           android:duration="500" />
fade_out.xml
<?xml version="1.0" encoding="utf-8"?>
<alpha xmlns:android="http://schemas.android.com/apk/res/android"
           android:interpolator="@android:anim/accelerate_interpolator"
           android:fromAlpha="1.0" android:toAlpha="0.0"
           android:fillAfter="true"
           android:duration="500" />
Use it in code like that: (Inside your Activity)
Intent i = new Intent(this, NewlyStartedActivity.class);
startActivity(i);
overridePendingTransition(R.anim.fade_in, R.anim.fade_out);
The above code will fade out the currently active Activity and fade in the newly started Activity resulting in a smooth transition.
UPDATE:
@Dan J pointed out that using the built in Android animations improves performance, which I indeed found to be the case after doing some testing. If you prefer working with the built in animations, use:
overridePendingTransition(android.R.anim.fade_in, android.R.anim.fade_out);
Notice me referencing android.R instead of R to access the resource id.
UPDATE: It is now common practice to perform transitions using the Transition class introduced in API level 19.