I.L.R what next after this

Hi
Can someone please tell me once my I.L.R has been granted what my next option is? i am not sure of citizenship or naturalization? i am an american citizen and don't wish to give up my passport but i have an ill father in the U.S.A and was wanting to go and visit him after my I.L.R has been granted as i have not left the U.K for over three years, BUT i don't want the hassle of all the questions when i come back to as or why i have been away for up to six months, is there another way i can do so when i go there and return back i don't have the worry of all the questions of why have you been away for some long? where is your partner? etc
any advice on the next steps would be great
thanks all

You are free to travel once you have ILR. You should not have any problems re-entering the UK.

ILR is lost if you are absent for 2 or more years.

Next step after ILR is naturalisation as British Citizen. You have to have ILR for a full 12 months and 5 years residency before you can apply for citizenship, unless you are married to a British Citizen and have 3 years residence and ILR, provided you meet the requirements and are within the absence limits.

USA allows dual nationality.

Also USA taxes you on all income inc worldwide, you might consider giving it up, I know lots who have...