Will Partitions and Indexes on the same table help in performace of Inserts and Selects?
May I suggest the following strategy:

Since there is a limit of 1024 partitions and sub partitions in single table have a Current 
table (1-2 months of data) partitioned by day and browser. And another History table for archive partitioned by month/day.

If you need to run reports on browser split, then have several tables with day/browser key.

Although there will be a number of tables (you may keep e.g. a table per quarter and then have second line of archiving to a table with month/browser key) it will be manageable, because partitions are quickly switched from one table to another.

Non-clustered indexes always help your reports, so having an index on a often used column 
is a good idea whatever partitioning strategy is implemented.

Depending on your reporting requirements you may also consider creating a summary table, which will have data already pre-aggregated for your reports. It can be populated overnight and speed up your reports dramatically (indexes is a must on this table too). You can even populate this summary table several times during the day without locking the table by using READ UNCOMMITTED - it is unlikely that you need precision to every visitor.